I think “non-insert” makes sense in the comment—overwrite mode is the alternative to insert mode.
I’m not sure if there’s a default style that’s unproblematic for everyone. As the vim mode shows, displaying a fat cursor that actually aligns with the character after the cursor is tricky, and I’d rather not pull in that complexity for this use case. Coloring the cursor is heavily theme-dependent, so again, not a trivial change.
As such, I prefer to leave this to integrations. Mentioning the class in the manual would be a good idea.