Rincewind, we are told, is a wizard. On the Disc, wizarding is a profession; Pratchett based them on the English academic system, with colleges and bursars and tenure. A wizard is a man of some academic distinction, or a student of such a man; career wizards are uniformly well-fed, of sound body (if not necessarily of sound mind) reasonably dressed, opinionated, crankish, and - importantly - capable of magic.
Rincewind is a wizard: he is not well fed, having spent his life being thrust from one adventure to the next; his body is more attuned for running away from things than it is for meandering the halls or sitting by a fire; his opinions largely revolve around “is this new thing going to eat me,” rather than more abstract matters; importantly, he is completely incapable of magic, in spite of years of study.
Rincewind is a wizard, and the interesting thing about that is that the reader is expected (and I certainly did) take both his and the narrator's insistence on it at face value. Why shouldn't we?
I had a conversation with @aeletich a while back, while she was teaching herself to program. I don't recall exactly what prompted it, but at one point I told her to stop worrying about all the better programmers out there: from everyone else's point of view, she was already a wizard. There might be better wizards, and worse wizards, but she'd already passed any sort of bright line delimiting “not a programmer” from “programmer.”
I think self-identification is important, and overlooked.