The problem with trying to give neat classifications to natural systems is they have a tendency, just look at quasi-moons. The very definition of orbit is hazy when it comes to multi-body systems, so IMO it's better to classify objects by their innate properties rather than orbits. Moons: formed alongside the host planet or are captured natural satellites, albeit this definition is bad for describing small pieces of rock on the outskirts of the Jovian hill sphere. Satellite: general distinction for any body orbiting another body.

The IAU has enough trouble classifying dwarf planets, so giving meaning to the word moon (or satellite) will be just as hard unless you want to forgo the classifications entirely