to run the AC
To me that does not look like the case here. the relay gets 12vdc from the AC demand output from heater controls. this give power to the relay and parallels to a output pin on the relay, the other output pin on the relay goes to the positive side of the clutch. the ground side of the clutch goes to the LP cut off switch. The ECM grounds the trigger on the relay thus supplying 12v to hot side of the clutch, if the LP cutoff switch is closed the compressor ground is made and the compressor runs, LP cutoff opens and the compressor ground in interrupted and compressor stops running.
The way I read it is the ECM has the ability to override the AC clutch on command( if it deems necessary) by switching the relay trigger (ground, brown wire) off this opens the relay thus dropping the 12v to the clutch positive side.