And if you can spot stuff I've forgotten about then you can judge me to be a different person. First or third person perspective - it's all the same. — Cuthbert

What I think, at this point of my journey, is that it is not at all the same.
Because only looking from our own viewpoint we are indifferent to what exaclty were the contents of the consciousness of that previous selves with which we still can identify ourselves. If I am comparing your current self with your self from last week, from first-person viewpoint, then I must not take in account the stuff which I spotted you have forgotten since last week. The only thing I must evaluate is - is your current self still aware of (some parts of) your last week's self. And if yes, then I'll have to judge that you are still able to identify yourself with your last week's self (i.e. "self sameness" holds), regardless of the fact, that I spotted some things which you have forgotten since. And if I am evaluating the personal identity of your both selves from third-person viewpoint (as my own self at last), I will have to take those spotted forgetten things in account and judge, that you are no longer the same person (i.e. "impersonal sameness" does not hold).

Now, is Mr. Locke's theory right or wrong - I don't know yet. I'll have to read his chapter XXVII a few times more, and I am still not sure if I will be certain after that. As I already told you I am in the very beginning of this journey. It is maybe possible that he was not making that first-person/third-person distinction clearly too. And it is maybe possible that Mr. Locke and Mr. Reid were both in this same trap in fact, with the difference that Mr. Reid felt uncomfortable and sensed something wrong.

I suspect you might be making Reid's point for him — Cuthbert

Could be, of course. But what I see for now is that "he who took the standard is the same person who was made a general" cannot be used in the same inference process together with transitivity, and it is precisely what Mr. Reid does. For if we want to use transitivity, then we have to judge from third-person viewpoint, and then he who took the standard is not the same person who was made a general. Mr. Reid definitely saw that something is not quite right, but I think he couldn't deliver it to us properly.

https://plato.stanford.edu/entries/identity-time/ - dense article, see the bit about Ship of Theseus — Cuthbert

I definitely will, thank you very much for the link.

And thank you so much for pointing out my mistakes in my clumsy attempts at reasoning, and for guiding me to a better position for my next attempt.

Seems there is still something that bothers me and I had to continue a bit further...
All of the following thoughts are based on a distinction between two different standpoints from which we evaluate the personal identity. On the one hand we can be looking at the persons we compare from the side as a third person, and on the other we can compare persons from within the subject's point of view. This distinction, of course, I did not contrive myself, I saw it mentioned by Mr. Simon Blackburn in his book for introduction to Philosophy "Think", in the very section dedicated to "the brave officer" argument. I did not pay enough attention to it the first time when I read it, and it's only after the @Cuthbert's insertion of "[my]" in his previous reply, when I (maybe) truly grasped it. In fact I only hope that I understood it, and it is quite possible that all of the following thoughts are plain wrong. But I have no other way to test that, except to post it all here and to wait for your judgements.

So, there it is:

I think the personal identity, as defined by Mr. Locke, regarded as a relation, has different properties when applied from the two different standpoints, mentioned above. Precisely, when we apply it from third person standpoint, then the relation is transitive, and when we apply it from the subject's standpoint, then it is not transitive. Therefore, I think, we have to consider them as two different relations.

For clarity and brevity only, in the following propositions I'll call the comparison which we apply as third person observers "impersonal sameness (of a rational being)", and the other one "self sameness (of a rational being)". For clarity again, when two persons A and B have "impersonal sameness", I will say that "A is the same person as B", and when they have "self sameness", I will say that "B identifies itself with A".

So, first, to claim "impersonal sameness" between A and B, in Mr. Locke's terms, we as objective third person observers need to know that the contents of their consciousnesses are the same. From this we can safely conclude, that "impersonal sameness" is a transitive relation, i. e. from "A is the same person as B" and "B is the same person as C", it follows that "A is the same person as C". In this case Mr. Reid's conclusion "that the general is the same person with him who was flogged at school" (i.e. "A is the same person as C") could be valid, if his previous conclusion "that he who took the standard is the same person who was made a general" (i.e. "B is the same person as C") was also valid. But "B is the same person as C" is not valid here, because as third person observers, we objectively see that B's consciousness is not the same as C's, for the former is aware of the flogging and the latter is not.

So when we regard the personal identity as "impersonal sameness", it seems that "the brave officer" argument cannot be valid.

Second, to claim "self sameness" between A and B, in Mr. Locke's terms again, we have to look from B's point of view, as a subject, and see that she/he/they are conscious of A's thoughts and actions (regardless of the contents of A's consciousness). In this case, when we regard the personal identity strictly as "self sameness", Mr. Reid's conclusion "that he who took the standard is the same person who was made a general" (i.e. "B is the same person as C") is valid. Although, I think, it would be more precise to put it as "he who was made a general identifies himself with him who took the standard" (i.e. "C identifies itself with B"). Alas, when comparing from the subject's point of view, we cannot jump freely to the next of Mr. Reid's conclusions "that the general is the same person with him who was flogged at school" (i.e. "A is the same person as C"), because the personal identity, when regarded as "self sameness", does not posses the property of transitivity. This can be proved by reducing to absurdity like this:
Let's suppose all the things that are supposed in "the brave officer" argument, also let's regard the personal identity as "self sameness", and let's suppose that it is a transitive relation. That would mean that if the soldier can identify himself with the flogged boy, and the general can identify himself with the soldier, then the general will be able to identify himself with the flogged boy. But that is a contradiction, because the general "had absolutely lost the consciousness of his flogging". Therefore the "self sameness" is not a transitive relation.

So when we regard the personal identity as "self sameness", it seems that one of the suppressed premises in "the brave officer" argument, precisely that the relation is transitive, is untrue.

It seems to me, that the only way in which "the brave officer" argument looks valid is when we are inconsistent in our standpoint and the two different personal identity relations are mixed in the same inference process.

That was it.
What do you think of it?

@Agent Smith, @Harry Hindu
Thank you for your answers. They definitely are a pleasant sources for further thoughts.

My concerns were of a bit different nature. I think @Cuthbert sensed very well where is exactly my problem.

I think I finally saw what I failed to see until now. It was probably the insertion of that

[my] — Cuthbert

into Mr. Locke's sentence that made me suddenly see the notion of "sameness of a rational being" from a different angle. Precisely to look at personal identity as personal. I was watching the contents of the consciousness of the "soldier" and that of the "general" and was wondering how we claim them same person, when those contents are different. What I failed to see is that Mr. Locke's theory allows the general, from his point of view, to identify himself with that person who took the standard, regardless of the contents of that person's consciousness.
I don't know if that is right or no. I also don't know why Mr. Reid assumes that personal identity is necessarily transitive. Most people would say it's "common sense", and claiming the opposite "makes no sense" and is "crazy", but what is common sense and what is crazy is a completely different subject.
In any case I now see that there exists a logical interpretation of Mr. Locke's theory that can render the "brave officer" argument completely valid, even if not sound for me yet.

I am unfortunately in the very beginning of my journey in Philosophy (and even more unfortunately this is not to imply that I am young), so I am not yet aware what exactly stipulation means in this context. I will of course try to fix this.
But if you are right, and whatever stipulation means, shouldn't then the conclusion sentence of Mr. Reid be like that:
"These things being supposed, it follows, from my own stipulation, that he who was flogged at school is the same person who took the standard, and that he who took the standard is the same person who was made a general"?
Why he uses "it follows, from Mr. Locke’s doctrine" instead, when it is now clear, that those things do not follow from Mr. Locke's doctrine, but from Mr. Reid's stipulation?

... we can see immediately that it makes no sense — Cuthbert

I am once again unfortunate, for it is not so obvious to me yet, but I am trying...

(1) Person A is the same as person B.
(2) If Locke's theory is right, person A is not the same as person B.

By modus tollens: Locke's theory is not right. — Cuthbert

If that was Mr. Reid's intention, and if we assume that (1) is an established fact, then I will have to accept his argument. But if (1) is not an established fact, then the "brave officer" argument, in this case, is merely displaying that Mr. Reid's own theory is not compatible with Mr. Locke's. Not much of an achievement, is it?

Probably it was phrases like "it follows, from Mr. Locke’s doctrine" and "consequence of this doctrine" that disoriented me, but I thought that Mr. Reid's intention was to prove that Mr. Locke's system of personal identity is not consistent in itself. And he decided to do it by demonstrating that Mr. Locke's conception of "sameness of a rational being" is transitive and at the same time not transitive relation. Which I failed to see as correct.

I think you say this because the one who took the standard remembers the flogging and the one who was made a general does not remember it. So the standard-taker and the bemedalled-general have different items in their consciousness. So they are (by Locke's theory) different people. Is that how you are arguing? — Cuthbert

Yes, almost. In place of different "people", they speak about different "persons".

The main point of Mr. Reid's objection is to show that by following Locke's theory it is possible to come to contradiction "that a man may be, and at the same time not be, the person that did a particular action". And what I see is that the Mr. Reid's conclusion "that he who took the standard is the same person who was made a general" is not valid, according to that theory. So, I think, that such contradiction as "the general is, and at the same time is not, the same person with him who was flogged at school" cannot be made.