In my airline, on my fleet, we use a derate and then usually a derated climb thrust too. There is nothing to stop us using a derate and full climb thrust though, or a full power take off (no derate or flex) and then derated climb power. Other fleets use flex and then reduced climb thrust as far as I know - so the same thing really.
I think this might be manufacturer naming conventions and nothing else. Certainly Boeing use Autothrottle and Airbus Autothrust in thier own documents, but that doesn't nescessarily have anything to do with lever movement.
Certain other things are named differently between the manufacturers with no discernible difference in function.