I must confess that I am not very familiar with Kant or that particular terminological distinction. Peirce, on the other hand, was very familiar with Kant; but he does not appear to have written anything about that particular terminological distinction. However, this comes from earlier in the same article:

Self-consciousness, as the term is here used, is to be distinguished both from consciousness generally, from the internal sense, and from pure apperception. Any cognition is a consciousness of the object as represented; by self-consciousness is meant a knowledge of ourselves. Not a mere feeling of subjective conditions of consciousness, but of our personal selves. Pure apperception is the self-assertion of THE ego; the self-consciousness here meant is the recognition of my private self. I know that I (not merely the I) exist. — CP 5.225, 1868

He seems to be saying that "self-consciousness" pertains to the transcendental ego as "knowledge of ourselves" or knowing that I exist, rather than the empirical ego as "consciousness of the object as represented" or knowing that the I exists.

I read the passage differently; Peirce seems to want to distinguish self-consciousness from both "pure apperception" which is the assertion of "THE ( transcendental) ego" [brackets mine], and also from the "internal sense" (which Kant calls the "inner sense", and which he understands to consist in the consciousness of oneself and one's private psychological states, or in other words, the empirical self).

As far as I know, there is no third option for Kant, and I cannot understand why Peirce seems to think that "recognition of my private self" is any different than recognition of the empirical self through inner sense.

"All experience implies a subject of experience" sounds a bit like

"I am able to see objects, therefore I must have eyeballs" — sime

None of what you say refutes or comes to terms with the issue of 'the subject of experience'. A deaf, dumb and blind subject remains a subject. And Robinson might be a singlularly un-self-aware subject, but he remains a subject nonetheless.

Peirce's take was that... — aletheist

What Peirce says is true enough but again I don't think it does justice to the question at hand. (Although I do recognize the distinction he makes in the second quote, between self-consciousness and pure perception. What I think he's reflecting on is the distinction of 'me thinking about myself', and the pure act of consciousness, which is not self-conscious but simply and choicelessly aware.)

As far as Kant is concerned, I think the key passage is that dealing with 'transcendental apperception'.

1. All experience is the succession of a variety of contents (pace Hume).
2. To be experienced at all, the successive data must be combined or held together in a unity for consciousness.
3. Unity of experience therefore implies a unity of self.
4. The unity of self is as much an object of experience as anything is.
5. Therefore, experience both of the self and its objects rests on acts of synthesis that, because they are the conditions of any experience, are not themselves experienced.
6. These prior syntheses are made possible by the categories. Categories allow us to synthesize the self and the objects.

5 is the key point - the 'conditions of experience' are themselves not objects of experience. This is in keeping with Kant's (and Husserl's) general notion of the transcendental as 'that which constitutes experience but is not given in experience'.

In practical terms, consider the question of self-knowledge, which is said to be at once of prime importance, but the absence of which is observable in oneself and in others. How could it be that we don't know ourselves? I think it's because underneath (or prior to) the level of discursive awareness, there are subconscious and unconscious factors that condition experience. Of course in Kant's day the Freudian terminology hadn't yet been devised but Kantian philosophy clearly anticipates it:

“The intellect remains so much excluded from the real resolutions and secret decisions of its own will that sometimes it can only get to know them, like those of a stranger, by spying out and taking it unawares: and it must surprise the will in the act of expressing itself, in order merely to discover its real intentions.”

WWR (World as Will and Representation) II(1844).

To relate this to the OP - the problem I see with panpsychism is the attempt to 'objectify' consciousness or to locate it as an attribute of external objects without fully grasping its elusive nature even within our own experience.

"All experience implies a subject of experience" sounds a bit like

"I am able to see objects, therefore I must have eyeballs"
— sime

None of what you say refutes or comes to terms with the issue of 'the subject of experience'. A deaf, dumb and blind subject remains a subject. And Robinson might be a singlularly un-self-aware subject, but he remains a subject nonetheless. — Wayfarer

Certainly it is true that from my perspective, my concept of a third person is of a subject of experience, or perhaps I might say, a potential subject of sensory stimulus. Likewise it is true that I see my empirical ego as a subject of experience, because of my tendency to think of myself as someone else situated in front of me.

But for me, here things must abruptly end. For this thought of the empirical ego as "someone else situated in front of me" is plain nonsense if it is intended to imply the existence of a literal hidden onlooker of my empirical ego.

Ordinarily, is it not the case that the notion of an onlooker or of an experiential subject is necessarily tied to thinking of a third party, whose status as a subject is in terms of behavioural stimulus-response criteria?

If so, then I cannot apply this concept to the personal pronoun in so far as it is used to mean something other than the empirical ego.

the "third-person" and "subject of experience" surely have identical uses, do they not?

If someone is doing the talking then someone has to be doing the listening. Maybe sometimes we do all of the talking, and none of the listening.

For this thought of the empirical ego as "someone else situated in front of me" is plain nonsense if it is intended to imply the existence of a literal hidden onlooker of my empirical ego. — sime

That is referred to as the 'homunculus fallacy'.

the "third-person" and "subject of experience" surely have identical uses, do they not? — sime

Clearly not. If I burn my hand, I don't say 'that hurt him'. I say 'that hurt me'. And that pain is a first-person experience, even though it can be described to some extent in third-person terms.

To relate this to the OP - the problem I see with panpsychism is the attempt to 'objectify' consciousness or to locate it as an attribute of external objects without fully grasping its elusive nature even within our own experience. — Wayfarer

If you only have access to your own personal experiences then that is solipsism.

Also you could not make the claim "Subjective experience has no objective existence" as you would lack sufficient access to any objective state to assert such a claim.

You might say "I believe there is no objective existence of experience" but by definition, there is no way to logically validate your belief as you have no access to the objective.

So the problem I see with your stance is that you try to state your claim as though it carries some weight of objective import.
It does not.

If you only have access to your own personal experiences then that is solipsism. — m-theory

First, it's not a matter of access. One doesn't open a door into personal experience or find it within some container. First-person experience is prior to any form of knowledge or inference (in this respect, Descartes' cogito argument is apodictic, i.e. cannot be reasonably doubted.)

What is it about experience that makes it 'personal'? What is 'mine alone'? In order for solipsism to be advocated, one has to take the additional step of appropriating the apodictic reality of first-person experience as 'mine'.

As regards knowledge of objects of experience, nowhere I have said that we don't have 'access' to knowledge of objects. What I've said is that knowledge of objects and the knowledge of the nature of experience are basically different, in that 'knowledge of experience' is first-person, that is, experiences are undergone by subjects, they are not objects in the way that bullets or billiard balls are. One can satisfactorily describe the nature of objects, in a scientific sense, without any reference to a first person, in fact, a large part of scientific method concentrates on doing exactly that. However one cannot describe the nature of experience in that way, because experience is not an object in that sense.

It is a modest claim.

For this thought of the empirical ego as "someone else situated in front of me" is plain nonsense if it is intended to imply the existence of a literal hidden onlooker of my empirical ego.
— sime

That is referred to as the 'homunculus fallacy'.

the "third-person" and "subject of experience" surely have identical uses, do they not?
— sime

Clearly not. If I burn my hand, I don't say 'that hurt him'. I say 'that hurt me'. And that pain is a first-person experience, even though it can be described to some extent in third-person terms. — Wayfarer

it sounds as though we're pretty much agreed then :) the transcendental ego, in so far as it designates anything at all, cannot be an onlooking homonulcus. i would simply like to further suggest that the meaning of all linguistic concepts concerns first-person understanding of third person behaviour. I might often use language to point at my private intuitions as it were, but none of this implies additional metaphysical substances to what is empirically experienced.

As for Kant's argument for the transcendental ego from the unity of apperception, he already appears to presuppose a subject that is distinct from its representations, the very thing he sets out to prove.

I suspect his belief in a transcendental ego is a consequence of his metaphysical understanding of time in terms of an atomic substance, albeit a mentalistic one. A deleuzean or whiteheadean process philosopher presumably has no need of a transcendental 'glue' to bind "temporally distinct" representations.