Is there a difference between "the rein back" and backing up a horse under saddle?

I use backing up as a punishment for Aya when she threatens to bite - it's the only thing that's really worked (I've tried slapping and it only makes her angrier, + head shy). She is sometimes a little jumpy just after being told off, but she doesn't seem any more inclined to back than previously unless she's thinking about biting, and then she'll be ready to back up if asked. I never use backing as a punishment under saddle, though.
@ClearDonkey - would Toofine one rein stop out of backing if you asked him?

What works really well for Dolly (not so much for Aya - better for some than others, I guess) when she barges past me, is yeilding her hindquarters. If your horse walks past you they put their hindquarters right near you, so you can just yield them in a circle and continue walking. Dolly figured out pretty fast that she wanted to keep her hindquarters back! It's the same as backing, though - you have to avoid overdoing it and make sure to yield them nicely plenty (not as a punishment, not aggressively, just gently so they don't start seeing it negatively).
Maybe that's why my horses aren't super inclined to back up, or scared of backing up, or anything, even though I sometimes use it as a punishment - I back them up nicely a lot, just if they're in the way, or whatever (which is pretty often).