Is there a better term for a pusher of this type ?

I mean a PUSHER means a person who deals drugs. Wouldn't for instance AXIAL THRUST JET ( ATJ ) sound a lot better ( or Axial Thrust Fan ) ?

Axial...because..the prop is connected to an axel that leads to engine unlike many other pusher.
Thrust...because the propfan produces propulsion/thrust
Jet...because this kinda plane goes at best solutions faster than regular jet ( LF 2100 was faster than a Citation I ) with less fuel ( Cri Cri with 36 kilos of jet thrust goes 150 mph...Max III will go faster with just 20-30 kilos of thrust provided by a 2-stroke piston engine ). Word fan is already used instead of the pusherprop...but also fan bears the similar meaning as the air circulator AKA fan...which lacks some certain appeal.

Jet engine propelled AC is actually quite primitive compared to this kinda ATJ lay-out ( even with a piston engine ) since it sucks lotsa air in and thus produce lot of drag as an airframe.

Word jet has been also used in jetsteam to describe a fast wind in high altitude also quite appropriately in an engine that we know as gas turbine.