No, I think your definition is the most common one (although in my experience nobody can agree on where to draw the line). Even after sleeping on it, I can't describe my feelings on the subject without sounding like a pretentious windbag :D Let's just say I prefer AU as a blanket term, but can accept AR as a subtype. So it can be more specific when it applies, but never more accurate.

In the case of "Pieces", I don't think it applies anyway. If it were as simple as "like the anime, except X happened instead of Y", I wouldn't have a problem summing it up :D I'd say it's a new universe built ouf of pieces (haha) of the original one. So it's superficially similar, but doesn't quite work in the same way.