We always referred to them as super/sub-scripts. So like xₙ is read “x sub n”

Upstairs/downstairs is kinda cute tho xD

Covariant and contravariant indices would be the formal terms. I'm not really sure whether I've seen "upstairs" written down.

Sub/superscript... strike me as the typographical terms, not the meaning? Like $x_\mathrm{alice}$ is certainly a subscript, and footnote 2 is a superscript, but neither is an index.