Face and Babyface are pretty much the same thing, face is just a short form for babyface.
Tweeners are in my view the most ridiculous creation of wrestling. A heel that gets cheered is a tweener? If he's getting cheered he's clearly doing something the audience wants him to do, something they'll enjoy, and babyfaces are the guys that do that.
Saying that Austin was a tweener when he was feuding with Bret Hart is mindbogling to me. Bret was being a crybaby, saying he had been screwed from the first moment he reappeared on WWE TV on the Fall of 96. Austin was just calling him on that, and the crowd agreeded with him, which lead him into babyface status. Hart's later actions (more crying and moaning, and prickish behaviour) just reinforced the notion that Bret was a heel, and Austin a face. A face with an attitude issue, but a face anyway, one of the good guys. The double turn worked so well because it was very organic.
Orton getting cheered when he feuded with Legacy was the bookers fault, since they wanted to have the Legacy turn face, but they booked their actions in a way that had an heelish feel to it. Would Batista's face turn had worked if he did something sneaky to eliminate HHH on the Elimination Chamber instead of the inverse?
If you book a face or a heel properly, there's no way that the crowd may turn him, Austin/Bret was clearly planned, Orton/Legacy wasn't. It's always booking.