No, I don't mean that at all. I meant nothing to do with memory. By 'identifiable', I mean 'able to recognise that something is a thing', distinct from the background field of reality. Babies don't need to have seen something previously, or to have any memory of it, for them to be able to identify things.
That's not what I meant, though. What I meant is that the speaker's use of a determiner allows you only to identify that it is a thing, not what type of thing it is. It's the word book that says what type of thing it is. Imagine that the hearer is blind. That wouldn't affect what the speaker says.In context 1 you can only identify the type of thing: it’s a book.
It really doesn't help that you insist on using the. I think this will be much clearer if we use this in this example.But the follow-up mental process is not part of the identification, but of getting new information (N gave me for my birthday) and of (semi-)unconscious deducing or inferring (I'm seeing this for the first time, but he has used “the” → N gave him only one book). But if I come to visit you with the same book again and say, "I've brought with me the book N gave me for my birthday" (the book I brought last time would be more natural, but we're discussing possibilities), it will be subject to identification in its true sense.
As I said above, it doesn't matter that the hearer is seeing the book for the first time (regardless of whether the speaker uses the or this).
The second time that the speaker brings the book would allow him a wider range of reference, because of shared knowledge. He could now refer to the book as, say, that book N gave me.
Let's forget about using the. I really don't think that's helping. Either that, or forget about using N gave me, which is confusing things even more. I mean, why would would a speaker use a defining clause in this situation anyway? If the listener is unaware of any books, I can't see any need for the speaker to say that this book is defined by the fact that N gave it to me. Are you sure you don't really mean to say the book, which N gave me? That would make more sense to me. Is that not what you mean by 'new information'?
May I ask you a question, just out of curiosity? If you had to choose one of the following ways to parse your noun phrase, which one would you choose?
1) the book N gave me
2) the book N gave me
Don't think about the grammar at all, just the way it makes most sense in you mind.
Is it not the case that it's mysterious because you invented it without having a good reason? Or do you mean it's mysterious in light of the fact that I said there may be more than one book that N gave me?That’s why I like this variant: it’s possible, but the reason is mysterious to me.
If the speaker uses a definite article and a defining clause, then yes, the hearer could very well imply that the speaker's point was to define the book as the single member of the class 'books N gave me.' I don't think my allowing there being other members of the class ought to trouble you.As well as “the book N gave me” removes openness, which should lead you to the conclusion that N gave me only one book.
But for some mysterious reason it doesn’t. At least you allow for more than one.

Student or Learner