Fair, I messed up the names there. Moonette is the synonym of that that I was thinking of.

But a satellite (natural or artificial) is any object that orbits around a celestial body. The Earth is a satellite just as the Moon is one. Subsatellite is just the satellite of a satellite, but that depends on context. Moons fit that definition, but aren’t usually considered subsatellites because we don’t usually think of planets as the satellites of the Sun that they are.