It's just a pass through or a relay, I wouldn't overthink and use any binary-electronics logic here.

If the pressure is above the min and below the max, voltage is carried to the compressor clutch. Two pins should suffice. Any "thinking" by the switch on whether to let the clutch engage goes on internally. If there was some requirement to provide an output as to WHY the compressor wasn't engaging, then you might need another pin. The compressor is dumb and doesn't care: on or off, +12 on the output side or not.