This has actually been a source of heated historiographical debate in the past. In the first half of the 20th century the popular hypothesis was that early modern witch trials were a response to a real phenomenon. The idea was that there was some sort of relic of pre-Christian Europe in the form of a nature cult which survived and whose adherents were persecuted as witches.

This idea fell out of fashion in the 70s and as far as I know serious historians now do not subscribe to it. This of course is very different to saying that nobody seriously believed they had magic powers. Belief in low level magic was very common in early modern Europe, and in fact a distinction was often made between 'malefica'- causing injury or nuisance via magic, and the more serious crime of consorting with the devil.

Also the lack of evidence for a Europe wide cult doesn't mean there weren't unusual religious practices around whose adherents may have believed they possessed special powers. The most commonly cited example is the 'Benandanti', who were a group of people in Italy who followed an esoteric visionary religion and were often persecuted as heretics and witches.

I'm travelling ATM so sadly can't check out my books and provide more information right now, so I'll try and come back later with sources and a bit more info. In general though, I'd say the answer to your question is that people definitely fairly frequently held genuine beliefs in their own magic powers. However, this doesn't mean that they subscribed to a consistent idea of a 'witch' that they considered themselves to be, and that the evidence doesn't suggest this was the case.

Edit- Right, back in front of my PC now, and can I recommend if you are interested in learning more about witches, particularly how the witch hunts interacted with other beliefs about the supernatural, checking out a guy called Julian Goodare. He has written quite a lot about the relationship between traditional Scottish folk beliefs about faeries, nature spirits etc and witchcraft. His work mostly focuses on Scotland, though he has recently published a big new book about European witch hunts in general. Must make a disclaimer though that the reason I know of him is mainly because his other work about Early Modern Scotland overlaps with my own, so I can't comment as to what degree of authority he is on witches and the supernatural, perhaps someone else might be able to shed more light on other good witchcraft scholarship.