Well, in an ideal world, I suppose that's how things would work. Parents would tell the stories to their children and adapt them as needed. The stories were part of an oral tradition anyways, not created to be written down. That's why there are so many different versions of the same basic story, because different people wrote the stories as they had been told to them.
But things are different today. If you sit down with a group of children and tell them you are going to read Cinderella or some other fairy tale, a good protion of them are bound to say they've already seen the movie. And if things are different, children are more likely to believe the movie is the "real" story than the book or a story told by someone.
Disney are simply the most well known versions, but not necissairly the best. It's important for kids to understand why there are a number of different versions, and the Disney movies are simply one interpretation.