Thermostat is the bi-metalic spring and flap with a probe attached to it. The handle you refer to is the primary air lever/ regulator. In my books. The terminology is a bit confusing when you are new to this. We have all been there.
Assuming if you have a digital thermostat in your home you don't call it a digital temperature adjust unit do you? If analog you don't say I'm going to adjust the analog dial or lever unit. It's thermostat. The bi-metallic spring and flap are a part of the thermostat as is the handle. The same goes for the air inlet on the back. If I adjust the air control down that is adjusting the thermostat down and vice versa.
But thanks for the nomenclature correction. That didn't answer my question from earlier.