You’re correct. In fact the largest desert in the world is actually extremely cold. It’s not the Sahara Desert, it’s the Antarctic Desert. A lack of precipitation defines a desert, not the temperature. The image you have in your head of the sand dunes, oases, mirages and a relentless sun are aspects of desert in hot climates. But those are not features that define all deserts.

You are also correct that the deciding factors for precipitation involve proximity to water sources, air currents, etc. If there is no water to evaporate to turn to rain nearby, or no air flow from water sources to be blown in, or if the temperature is so low that the humidity remains essentially zero at all times, then little to no precipitation can happen in a region. That’s a desert.