Yeah I've seen that :) but what i mean is how quickly from the fridge compressor kicking in does it start to cool?
The big cooling lag would generally be, the time taken for a container of liquid to reach the surrounding air temperature, rather than time taken for air in fridge (or freezer) to reach the set temperature.
The former, will reduce, as the differential temperature increases (ie the colder the surroundings).
A flat bottom vessel, sitting on a cooling element would react the quickest. Fridges (& freezers) that circulate air, will cool the liquid quicker, than those that rely on convection (think wind chill factor).
As Antigozo pointed out, the temperature inside the fermenter goes up during active fermentation. This is because fermentation is exothermic, at that stage. And it's temperature can typically increase to 2 to 4°C above ambient.
A lower temperature setting might be recommended for the first few days, so that the exothermically increased temperature, doesn't exceed a good temperature for the yeast.
If say, the desired temperature is 20°C, the fridge is controlled to cool to 19C, and the heater to 21C. During vigorous fermentation, more heat is being generated, than would be removed at the 1°C difference (between set 19 and desired 20), so temperature will increase.
Having the temperature probe inside the fermenter, should give a big improvement on the above. Though I'm not sure, that I'd trust a regular inkbird probe to be waterproof.
But vertical position of probe, within the vessel, can still cause a fairly big difference, due to temperature stratification. As during less vigorous fermentation, the heated liquid from heater tube (usually towards top), being lighter stays around the top. While liquid, cooled by contact with the base and walls, being heavier sinks to the base.
I'd be interested, in seeing fridge stratification figures, if anyone's done any tests.
I don't have a fridge system. But using a heatpad, in a cold garage, and under lots of bubble wrap, the temperature (vessel exterior wall) can be over 6°C cooler towards top (of liquid). Despite the helpful convection currents, in this situation.