In The Land Of Johnsons (yeah, I went there
) "cut-in" versus "cut-out" comes down to which side of your chosen Set Point you want the cooler or heater to run.
For all four examples, assume Set Point is 40°F and Differential is 5°F
In Cooling/Cut-Out, the compressor will come on at Set Point + Differential (45°F) and shut off when the temperature drops to Set-Point (40°F).
In Cooling/Cut-In, the compressor will come on at Set Point (40°F) and shut off when the temperature drops to Set-Point - Differential (35°F).
In Heating/Cut-Out, the heater will come on at Set Point - Differential (35°F) and shut off when the temperature rises to Set-Point (40°F).
In Heating/Cut-In, the heater will come on at Set Point (40°F) and shut off when the temperature rises to Set-Point + Differential (45°F).
Cooling/Cut-Out and Heating/Cut-Out make the most sense to me and are the paradigms that most controllers use.
The two Cut-In modes are not only superfluous, they just confuse folks...
Cheers!