The first thing I check when I have a noisy AC compressor is the belt tension. On several cars I've worked on tightening the belt correctly got rid of the noise. I didn't believe it till I tried. In your case though I have doubts this is the problem because yours is noisy when it's off.
I believe your car has a serpentine belt so the belt tensioner should be checked. I believe the tension is set by aiming the pointer correctly and not by measuring the belt tension directly.
Check the AC idler pulley if you have one. I once had one go bad on me and replacing it got rid of the noise.
Someone else with more experience could probably tell you if the AC clutch is the most likely suspect but checking belt tension and AC idler is easy and cheap.
1982 300D Turbo