I don't usually bother scoring advances on throws at all. I mean, it's not a hit, it's not an error, so why bother even keeping track of it?
Technically I think it's a species of fielder's choice, FWIW.
As for this:
"If the runner is standing on the bag, sees the throw go to the wrong place so he advances, then it could be scored an error on the throw."
I don't think that's typically done unless the ball gets loose and the runner's advance is caused by that. That sort of error in judgment -- e.g., throwing to home when it's really too late to have a play there so that the guy who hit a single can take second base -- is not scored as an error.
There are many things that the defense can do to prevent runs that are not errors. That's why the rules for determining ERA do not do full justice to a pitcher with a bad defensive team behind him.