I don't think most "small charter operations" would be General Aviation (unless I understand what's being chartered here?) and although most GA planes are pistons that's not part of the definition. What matters is
why you're flying, not what you fly.
The categories (for non-military use) are generally Scheduled or Air Transport (any time you buy tickets for a flight, that's the category, you don't know or care who is flying, you paid for the journey between a specific origin and destination at a specific time; a FedEx plane is also Transport), then Commercial (not Transport but somebody is getting paid to fly aircraft, maybe it's crop spraying, TV news copter, police, or just another TV priest being flown around in his private jet), and only if nobody was getting paid is it General Aviation.
If your cosmetic dentist can afford a brand new Vision Jet so that he can live 100 miles away and fly in to do $5000 appointments without sitting in traffic, that's General Aviation. The authorities don't care that he's getting paid to be a dentist, he's not getting paid to fly his plane.
If your airline uses a relatively tiny PA-42 to get customers to an obscure but important airstrip with maybe 3-4 passengers per day that's still Air Transport.
On the other hand if some oil sheik owns their own A320 with their own custom decor and has a team of pilots to fly it wherever they want, that's still only Commercial, not Air Transport because nobody is buying tickets, it just goes wherever he wants.