Best I can tell; The electric clutch is not engaged in either of those videos. It is easy to see on the Crossfire with the hood open from the top, so double check as your video is hard to see. The outer pulley has the grooves and the belt will always keep that part spinning when the engine is running. With the A/C off the inner part (electronic magnet) will not spin. It can be seen as it extends out towards the front of the vehicle. On the Crossfire it has three rivets (they are silver on my car) that are easily seen visually. The extended part with the rivets is attached to the compressor shaft and should only spin when the A/C switch is turned on and it should not spin when the A/C switch is off.

Yes, when you have the hood up check fuses #15,33,36 and 37. Best if you have a volt/ohm meter, but visually at the least.