> I guess you could try to claim that only the way HVAC's move heat around qualifies for being called a heat pump, not any other way of moving it should count right?

Anything that moves heat from one area to another, not just evening them out but actually forcing the heat to move, counts as a heat pump. HVAC style, or peltier style, or other methods with tension or chemical reactions all count.

A combination heater and circulator does not do that. It is not a heat pump.