In what way do magical powers define witches and wizards?

I'm currently re-reading HBP and a question has begun to form that has been nagging on the back of my mind for ages. In very simple words that question is: What does magic really mean for witches and wizards? How integral a part of their selves is it?

Basically, witches and wizards define themselves over magic. Magic is part of what they are. It's not just an extra skill but rather an inherent power that, carefully cultivated and honed, results in extra skills that are used to faciliate many aspects of life. Right? Magical powers are a bit like intelligence: a gift of nature based on which extra skills can be acquired. This would make witches and wizards the prodigies of the human species, and the way many witches and wizards talk about Muggles, it is apparent that this is what they consider themselves when compared to Muggles.

( So what do magical powers and the loss thereof mean for a witch's or wizard's sense of self? )