What is the linguistic purpose of 'one' as an adjective preceding a noun?

It is a determinar used to add emphasis:

One can be used instead of 'a' to emphasize the following noun. [emphasis]

There is one thing I would like to know–What is it about Tim that you find so irresistible? One person I hate is Russ.

(Collins Dictionary)

'remember when we saw that one guy?'

I interpret this as meaning "unique". For example, suppose we attended an entertainment event and there were ten players and one of them was wearing a bear's head. Later we might say, "Remember when we saw that one guy - the one with the bear's head?"

'my one exam just got postponed'

I interpret this as meaning "one-and-only". I had only one exam and it was postponed.