But, for some conversations, we can use modal logic and take the metaphysically actual world as the modally actual world, and look that the accessibility relations that originate in the metaphysically actual world. — Banno

We cannot "take the metaphysically actual world as the modally actual world" because the difference between these two is the difference you insisted that we must respect. The "metaphysically actual world" is the world we live and act in. The "modally actual world" is a representation. That is the difference which you accepted in the other thread, and agreed that we must respect. By the same principle, "Nixon" refers to something different in the metaphysically actual world, from what it refers to in the modally actual world. And we must respect the fact that there is a significant difference between these two, to allow for the reality of incomplete, mistaken, or otherwise misguided representations posing as the "modally actual world". As you say, the modally actual world is just another possible world, though it is assigned special status.

This is why I emphasized in the other thread that truth is a judgement. The representation is judged to be adequate, and given the name "actual world", but it is still just a representation which could be mistaken. But "truth" does not mean correspondence in modal logic. It is very important to respect this difference between the representation within the modal model which is called "the actual world", and the real "metaphysically actual world", because "the actual world" in modal logic can be created from a variety of different principles which do not necessarily require rigorous criteria of "truth" in the sense of correspondence. The "modally actual world" does not necessarily correspond with the metaphysically actual world. That's the deficiency of assigning "truth" an entirely extensional meaning, which the article refers to in section 1.2. Truth is arbitrary. This is required to make the modal model effective. Instead of an intensional criteria for "truth", there is an extensional stipulation.

Possible world semantics, therefore, explains the intensionality of modal logic by revealing that the syntax of the modal operators prevents an adequate expression of the meanings of the sentences in which they occur. Spelled out as possible world truth conditions, those meanings can be expressed in a wholly extensional fashion. — SEP

Another way to think of a intension is the rule we apply in order to decide, say, if that bird is a swan or not. But the truth of "That bird is a swan" is completely determined by the extension of "That bird" and the extension of "...is a swan": it will be true if and only if "That bird" satisfies "...is a swan" — Banno

This is a good example, I suggest you take a good close look. With the intensional definition we have criteria, "the rule" by which we judge whether or not "that bird is a swan" is true. We follow the rule and make the judgement. By the extensional definition however, "that bird is a swan" is true if that bird is a swan, i.e. is a member of that set. In this case the judgement may be completely arbitrary. Without an intensional definition, we can decide for whatever reason we want, whether or not the bird is a swan, we place it in the set of swans or not, and this forms the grounds for whether or not the proposition is true.

However, as Plato showed, Pythagoras avoided that arbitrariness by assuming real independent Ideas, and the theory of participation. The independent Idea serves as the criteria for "swan" which human beings don't necessarily know, making the bird a member of the set or not, without any human being needing to judge. This is Platonism. Now there is an eternal objective Idea of "swan", and it is true that all the birds who are swans, are swans because they partake in this Idea. So "is a swan" is satisfied if the bird partakes in the Idea of swan, whether or not a human being makes that judgement. Furthermore, "'that bird is a swan' is true if that bird is a swan", implies that the latter "is a swan" means partakes in the Platonic Idea of swan.

We cannot "take the metaphysically actual world as the modally actual world" because the difference between these two is the difference you insisted that we must respect. — Metaphysician Undercover

All I did was point to the difference between metaphysics and modality.

And this is not my account. The account here is the standard account of logicians.

But you twist and swivel.

There are things that you could say here that would be interesting. But your inability to understand modal logic prevents you from framing them in anything like a coherent fashion.

Step by step.

1. The core mistake: reifying the “modally actual world”
Your opening move is this: We cannot "take the metaphysically actual world as the modally actual world" because … the "modally actual world" is a representation. This misfires because in possible-world semantics, “the modally actual world” is not a representation of the metaphysically actual world. It just is the world designated by the model as actual. There is no further ontological claim being made.

As I have said, within a modal model, we stipulate a world as actual, and then examine accessibility relations from it. That stipulation does not compete with metaphysical actuality; it is a modelling device.
You are treating the model as if it were trying — and possibly failing — to represent reality. But modal semantics is not representational in that sense. It is instrumental. So the objection attacks a position that isn’t there.

2. Confusion between semantic stipulation and epistemic judgement
You write "This is why I emphasized … that truth is a judgement". That is false, or at least badly equivocal. In modal semantics, truth-at-a-world is not a judgement, nor is it an epistemic act. It is a semantic relation defined by the model. No one is “judging” that Nixon exists at a world; the valuation function assigns extensions at that world. That’s it. You slide illicitly from truth-in-a-model to truth-as-human-judgement. This is a category mistake.

You are psychologising semantics.

3. The Nixon move fails for the same reason. You say that "Nixon" refers to something different in the metaphysically actual world, from what it refers to in the modally actual world. Again: no. Within a model, “Nixon” has an extension at each world in which it exosts. Across models, reference is fixed by interpretation. None of this implies that the model’s Nixon is a representation that might be mistaken.
Mistake only arises if you assume the model is making a claim about the world. It isn’t. It’s a tool.
This is exactly the point Kripke, Lewis, and the SEP article are making — and which you are resisting by importing epistemology where it does not belong.

4. Misreading SEP on extensionality
You quote SEP as spelled out as possible world truth conditions, those meanings can be expressed in a wholly extensional fashion, then respond that truth is arbitrary. That is simply incorrect.
Truth is not arbitrary; it is stipulated relative to a model. That is not arbitrariness in the philosophical sense, any more than choosing a coordinate system is arbitrary in physics. Extensionality ≠ lack of constraint. Instead, once the model is fixed, truth values follow mechanically.

You are conflating “not grounded in metaphysical correspondence” with “arbitrary”. Those are very different claims.

5. The swan example: a serious error
You write that without an intensional definition, we can decide for whatever reason we want … whether or not the bird is a swan. This is flatly false. In extensional semantics, membership is fixed by the interpretation function. There is no discretion left to the user once the model is set up. You are smuggling human judgement back in again, where it explicitly does not belong.

Intensions explain how extensions vary across worlds, not that extensions are chosen on a whim.

6. The Platonic turn is a non sequitur
Your appeal to Plato and Ideas does no work here. Possible-world semantics is neutral on whether universals are Platonic, Aristotelian, nominalist, or fictional. Introducing Forms does not “solve” a problem — because there was no problem to begin with. You move from “extensions are stipulated in a model” to “therefore we need eternal Ideas” That inference is invalid.

Modal logic does not require metaphysical grounding to function, any more than arithmetic requires Platonism to be usable.

You are repeatedly:

• mistaking semantic machinery for metaphysical representation
• mistaking stipulation for arbitrariness
• importing epistemology into model theory
• and then trying to fix the resulting pseudo-problem with Platonism

The critique dissolves once the role of possible-world semantics is properly understood.

Platonism is common in mathematical interpretations. The Idea of "two" for example, is supposed to have real meaning, independent from human minds, so the symbol stands for that intensional package of meaning, as an object. — Metaphysician Undercover

As you say, an extensional definition is trapped by limited empirical observations.

Suppose there is an extensional definition of S, where S = {two red books, two green trees, two black thoughts}

You say that the elements of the set S are there because they are part of the Platonic “Idea” of S, where the “Idea” is the intensional meaning of S, and is objective and independent of the human mind.

The problem is, how can you go from the extensional definition of S existing in your mind to an intensional definition of S existing independently of your mind?

IE, what is the intensional definition of S?

I think the same sorts of questions could be asked about ordinary language use. If Paul talks about x, what guarantees that Paul knows what his words mean?

Btw, this:

The Idea of "two" for example, is supposed to have real meaning, independent from human minds, — Metaphysician Undercover

is straight up incorrect. This is the type of thing Meta just pulls out of his butt.