"...Face, does the starter engage and actually try to crank the engine, or does it merely sit and click?..."
Hmmm...It just clicks, now that I think of it. I tried to start it by popping the clutch while it was in a roll, but it wasn't a steep enough hill, I'm sure.
Sounds like the starter.
Sounds like what happened with the 1990 Geo Metro where the starter brushes arc cut a gouge into the starter commutator thingy and would only crank at certain starter positions.
(Sometimes letting it sit a bit would get it to start.)