I know the chord stays Em or E major or whatever, wherever the notes are. I was wondering if there was any simple convention for identifying these up-the-neck chords with open strings - and it sounds like there isn't.

Not having names for them makes it harder to pass them around. For example, you can't really say "try playing that E major chord where you play an A shape as if you had a capo on the seventh fret, but you only finger the third fourth and fifth strings"