Player Two: I don't know and I don't care. Bring it on, I just play good moves and look for tactics.
I have a friend who knew slightly more than that, but not much. He certainly didn't know the names of different types of open Sicilians. He literally said "I usually just sac something on e6 and it usually works out". He was 2150 at the time.
Sooner or later the opening leads to a unique position that isn't covered in a book. That's when player one's advantage slowly goes to naught. Sometimes a change in the pawn structure is more than enough to leave someone lost.