> They use a different cycle with the refrigerant.

No, every major consumer air conditioner and every major consumer heat pump on the market that I'm aware of uses the same cycle (the vapor compression cycle). The only difference between a vapor compression cycle AC and a vapor compression cycle heat pump is which of the two loops is being used as a heat sink and which as a heat source. You could in theory do this by simply reversing the flow direction in the refrigerant loop, but in practice the better approach is to do the same thing with valves, allowing the compressor to always operate with the same flow direction.

It's important to distinguish between the heat pump itself and the heat exchange medium used to transport that heat into a given space though. If you try to connect a heat pump meant to replace a radiator system in an old (or european) dwelling, the radiators we're really designed for cooling, and you may have performance issues. But that has nothing to do with the heat pump itself.

How so? Anything using vapor compression refrigeration should technically qualify as a heat pump. They may not move the heat as far, but that shouldn't matter. A swamp cooler isn't a heat pump, but the units you describe would seem to me to obviously qualify.

You can buy portable window heat pumps from several companies. They’ve been around for a while.