I've only been there once. (OALD)
Does this mean ‘I’ve just been there not doing particular activities like studying or staying for particular something else, and I have once? (Because ‘only’ is put between verbs, it is different from ‘I’ve been there only once’.)
He has only seen her once. (The Cambridge Grammar of the English Language)
Likewise, does this mean ‘he has just seen her not talking to each other et cetera, and he has once’?
Actually it's not at all different from "I've been there only once." The two sentences have the exact same meaning. "I've only been there once" is certainly more idiomatic, and you might be able to argue that it emphasizes the only a bit more since it appears nearer the beginning of the sentence. But there is no difference in meaning.
To elaborate on the possible small distinction in emphasis that could be made:
I've been there only once.
The statement is that you have indeed been there; but you're adding that you've only been there one time.
I've only been there once.
You're leading with the only, emphasizing that your being there has only happened one time.
But this is a very slight distinction, and the meaning (that you've been there one single time) is exactly the same no matter which way you phrase it. I think "I've only been there once" is more idiomatic, but beyond that it's your choice which you use.
In that specific case, there isn't much difference (or no difference at all) between "I've only been there once." and "I've been there only once." That is not always the case, though.
For example, the following sentences have different meanings.
Only I was waiting for her.
I was only waiting for her.
I was waiting only for her.
The first sentence means there was nobody but me waiting for her; the second one means I was not doing anything else than waiting for her, or that was my only reason for staying there; the third one means I was waiting for her and not other people.
Being in a place doesn't mean you are not doing anything, or you are just waiting; for example, if I say "I have been at work since 6 AM." I don't mean I didn't do anything, or I was waiting for some event to happen. If you mean that you were doing something, and only that, you need to explicitly say that.