Thanks. Yes, I agree. Ordinary usage outside the classrooms and the definitions in other dictionaries (OED for instance) are much better.
The problem is that this makes the schoolkids even *more* confused. Their parents and most stuff they read from reputable sources use ‘fact’ in a way that, according to their official teaching, is incorrect. For instance, we often implicitly reason that X is not a fact because X is false. Schoolkids can’t follow this reasoning; what they’ve been taught entails that that inference is invalid.