Excuse my lack of an actual answer, but life is too short to try and understand syllogistic logic through a translated 2000 year old text.

Instead, have recourse to something like this: http://intrologic.stanford.edu/materials/logicinaction/ch3.pdf

I appreciate the resource. I do wish it had a little more clarity on this exact issue though. I am less interested in the absoluteness of ideas behind the syllogism, I know I can crack open any modern logic textbook for that. I'm more interested in understanding the history of thought behind the ideas and tracing that progression. So instead of trying to understand this from the perspective of "how do these syllogisms work?" I'm approaching it more from "What must Aristotle in particular have been thinking when he wrote this?" angle. To be honest, I'm a little astonished that this kind of stuff isn't easier to find discussion on. The internet is young, sure, but Aristotle is Aristotle, I'd think there'd be more exhaustive discussions behind the nitty gritty of his works than most other philosophers. And I know I'm not very smart, but I can't be the only one who has gotten tripped up here and elsewhere at parts where I found my confusion to be high but discussion to be lacking

You are right to think that this text has been studied carefully. I have a read a lot of Aristotle but this work is not one I have devoted time to. I will look around for attempts to unravel the text.

The internet is young, sure, but Aristotle is Aristotle, I'd think there'd be more exhaustive discussions behind the nitty gritty of his works than most other philosophers — Pretty

There should be some reading guides written by experts, possibly even for the specific work you are struggling with. I would try to download some of those pdfs and go through them.

Give a go to Commentary on Aristotle, Prior Analytics (Book II): Critical Edition with Introduction and Translation by Leon Magentenos. Maybe you are looking for something like this:

Superb, thank you so much. I'll look into this now and see if it can resolve my confusion. I swear I googled for something like this a million times and nothing came up, so I appreciate you

Let us know if you come to conclusions.

Hi friends, I'm back and centered in on the precise part of this commentary that discusses the part I'm confused on. This commentary does somewhat enlighten things with good examples at every step, but I'm afraid it doesn't really relieve the source of my confusion. Here is the excerpt of the relevant passages and my interpretations on each:

7. [...] There is, however, yet another cause for which universal syllogisms infer plenty of conclusions. For from the content of the minor term you will draw a conclusion and from the content of the middle term another one, but all conclusions will be drawn in the first figure through this very figure, the first one, by means of which the first syllogism also comes about. Let us assume that A stands for essence, C for animate being, E for sense‐perceptible being, B for logical being and D for human being. The first syllogism is that essence is predicated of every animate being, animate being is predicated of every logical being, and therefore essence is predicated of every logical being. Indeed, this is the first syllogism. Since the minor term, the logical being, contains the human being, you will draw from the content of the former a second conclusion: essence is predicated of every logical being, logical being is predicated of every human being, therefore essence is predicated of every human being. But even the middle term, the animate being, contains the sense‐perceptible being and you will draw another conclusion: essence is predicated of every animate being, animate being is predicated of every sense perceptible being, and therefore essence is predicated of every sense‐perceptible being. [...]

I've abridged this chapter a bit where he reviews the previous chapters in the beginning, and where he goes over the same idea but as a negative syllogism at the end. However this is right where the confusion arises, right at the beginning of the discussion of subordinates. On its own however, there is no contradiction yet. Magentenos here specifies that the extra subordinate conclusions that you can get from a universal-positive first-figure syllogism, also use universal-positive first-figure syllogisms. This he elaborates by laying out his terms, giving a first figure syllogism, and then deducing the subordinates of both the conclusion (aka the minor term) and the middle term, respectively. This he does, as already said, by means of the first figure unto these subordinate terms, as well as being in some way branched off of the original syllogism of the first figure. I have italicized the last syllogism because it will be the critical part of the contradiction to come: note carefully how its premises do not include the original conclusion AB-- this will be the card we pull back out at the end of this comment.

Here is a diagram:

8. In the second and in the third figure, not all conclusions are drawn by the same figure, but by a different one on each occasion; e.g. in the second figure it was inferred through the middle term living being that stone is predicated of no human being. The particular term, namely grammarian, which is subordinated to the minor term, namely human being, will be inferred in the first figure as follows: stone is predicated of no human being, human being is predicated of every grammarian, therefore stone is also predicated of no grammarian. Similarly, logical being, i.e. the particular term subordinated to the middle term, namely living being, will also be inferred in the first figure as follows: stone is predicated of no living being (for ‘no’ converts to itself), living being is predicated of every logical being, therefore stone is also predicated of no logical being.

Here we bring in the second figure, where the more of the confusion lies, but still no contradiction yet between these two chapters. First Magentenos makes explicit that in this figure, extra conclusions will be drawn using figures other than the one they're branching from. That is, from the second figure, other figures like the first will be utilized to make further conclusions. He then shows this. He makes an allusion ("it was inferred") to a second figure syllogism using terms Living Being, Stone, and Human Being (I cannot find an actual previous reference to a syllogism with these terms anywhere, except for later on in Prior Analytics when discussing false premises.). Then he again substitutes subordinates for the conclusion and middle respectively, to make further conclusions. As he lays these two syllogisms out, he notes how each are deduced using the first figure.

He also, oh so conveniently, swaps around the term letterings from here on without saying so (perhaps related to the mysterious allusion?), so here they are in the second figures diagram:

9. In the case of the second figure, however, a syllogism will be possible only for what is subordinated to the conclusion, namely for the subordinated term. A syllogism for what is subordinated to the conclusion will be possible from a premise demonstrated by syllogism; for the major premise ‘stone is predicated of no human being’ was the conclusion of the second figure and every conclusion has been demonstrated by a syllogism.

This is a crucial part. The first sentence is the first part where a contradiction seems apparent, and it comes up in the same way in the Aristotle text. This can be accepted on its own if you take it on its own, but with what we established so far, it doesn't, as we shall see in full. The second sentence in this chapter explains the reasoning for this sentence: the syllogism is possible for the conclusion, because it will include "A premise demonstrated by syllogism" , that is, it uses a premise that was previously a conclusion in the figure it branched from. The contradiction seems like it comes from a sudden shift in topic. It feels like it confuses itself between the formerly established idea of extra conclusions that use the same/different figure as the ones they branch from, and the here newly established idea extra conclusions that, regardless of figure, use the conclusion from a previous syllogism as one of its premises.. But why this sudden change in topic, and why is it specified for the second figure, when as we shall see it applies to the first figure above as well?

10. To the terms subordinated to the middle term; or rather the conclusion of the terms which are subordinated to the middle term, namely to A, that stone is predicated of no logical being, has not become clear by means of a syllogism; or rather, the major premise, that ‘stone is predicated of no living being’, has not been demonstrated before, but it was received undemonstrated. Furthermore B, namely the stone, is predicated of no E, namely of no logical being, because logical being is subordinated to A; and since B is denied of A, namely of the living being, it is will also be denied of E, namely of the logical being.

And this is where the contradiction is finally made explicit. Aristotle and Magentenos both say here that the middle subordinate is not concluded by means of syllogism, specifically because neither of the involved premises come as conclusions from previous syllogisms. Let's pull our card back out from the top of the comment, the bolded and italicized part, and remember that this has already taken place in the first figure, with no mention. When Essence was predicated of the middle term Animate Being, and then Animate Being was in turn predicated of its subordinate sense-preceptive being, this was treated as using the first figure deduction. But in the second figure middle subordinate, Where Stone is negative-predicated of Living Being, and Living Being is predicated of Logical Being (see chapter 8), it is considered not through the syllogism because neither of those predications came to our disposal through a deduction. And yet, neither of the predications in the first figure example come to us through a deduction either-- they were both granted to us. This contradiction seems to come from the confusing, perhaps accidental, change of topic that I mentioned above. Aristotle seems to begin by talking about the figure-structure of syllogisms that branch off of a given figure and provide extra conclusions. He then appears to begin to give conditional statements to the second figure with regards to this topic, by moving on to speak of the second figure alone, but as he does this he actually changes topics altogether, to focus on the nature and source of the premises used to find extra conclusions, regardless of figure, and perhaps even regardless of deductive status of the conclusions. This final chapter/passage of Magentenos below attempts briefly, but not adequately, to explain things.

11. That B is predicated of no C, however, was demonstrated by a syllogism of the second figure and is conclusion of the latter; it was taken as major premise leading to the conclusion that ‘stone is predicated of no grammarian’. But that B is predicated of no A, which is exactly the major premise leading to the conclusion that stone is predicated of no E, namely of no logical being, is undemonstrated. Consequently, that B is predicated of no E was not inferred because of the syllogism, or rather it does not result from a premise demonstrated by means of syllogism.

This is just rephrasing what we've all been over. First he restates that we got a certain conclusion through the second figure's standard process. Then he says that this conclusion is used in turn as a premise in the deduction concerning the conclusion's subordinate. And then he says once again how this does not happen for the subordinate of the middle, and so that subordinate should not be considered as demonstrated through syllogism. This just seems to further show definitively that we have abandoned the discussion of figure-structure that took up chapters 7 and 8 of Magentenos, and have totally started seeing things in terms of the source of various premises.

I can't tell now who is the more confused between myself and Magentenos. Going directly back to the Aristotle passage, at the beginning he does specify: "all the things that are subordinate to the middle term or to the conclusion may be proved by the same deduction" ... but "proved by the same deduction" is the vague term in question. Magentenos seems to interpret it in his chap 7 and chap 8 as "proved with the same figure" and thus elaborates how the figure stays the same. But if we want to assume the ever-so-smarter Aristotle was consistent, then we should probably take it to mean "proved by using propositions from a previous syllogism as premises" that is, the subordinates will source at least some of their premises from the previously established syllogism (and by 'proposition' I mean either premise or conclusion). If we accept this, then instead of Magentenos' interpretation of sporadically switching from figure-structures to premise-sources, we could interpret Aristotle's original passage as broadly talking about the sources of premises in both parts. However, this still doesn't alleviate the contradiction, because even if we do interpret "proved by the same deduction" as "proved by using propositions of a previous syllogism as premises" it's not the same as "proved by using the conclusion of a previous syllogism as premises" which is what both Aristotle and Magentenos clearly focus their attention on by the end. However if we still grant, as I really wish to, that Aristotle did stay consistently on topic, since I can't find any smarter person supporting the idea that he didn't, then we still don't have an explanation for why he didn't point out that the first figure's middle subordinate also doesn't utilize any previously deduced conclusions. Like I said in a previous comment, I'm less concerned about the actual truth of the syllogism and more concerned about the seeming error in the text and the lack of historical attention brought to said error. How have countless students in the past not torn their hair out at this exact passage, enough to warrant plenty of provisions and footnotes as warning?

Thanks to everyone who has helped me so far and has bothered to trudge through this comment to help me. I will continue as always to poorly search for further commentaries but I would appreciate any more that anyone else might find!