One advantage of a freezer over a fridge is that, with the door on top, all the cold air doesn't spill out when you open it. When my fridge (fermentation chamber) is opened, yeah, it kicks on. Freezers (lagering and keezer), no. Anyway, how often are you going to open it? The difference in how much work your compressor does should be negligible. Also, the jar of water isn't a real analog for the beer for those using the freezer (or fridge) as a fermentation chamber, as it isn't generating heat like the ferment. Just another consideration.
I think if I had the probe in water, I'd want zero "swing" set into the system, because by the time the probe registered a change, the beer would already have changed temperature. A few degrees swing in air temperature, and the beer will stay steady.