The plate is the thing that gets pulled into the clutch by the magnet. It is splined to the compressor shaft and fastened with that little bolt in the center. If you can stabilize the plate, say with a screwdriver wedged between a couple of those 3 outer bolts, you can tighten or loosen the center bolt on the shaft. If you remove the center bolt, you should be able to pull the plate off the shaft by hand, allowing you to see the clutch assembly.

The AC clutch assembly, with the pulley, is pressed onto the snout of the compressor housing. If the pulley is moving around as you described, it sounds like the clutch assembly has come off of the housing somehow, or its bearing got really bad. It does not take an enormous effort to press on or off, but you do need to get at it from certain positions. Most likely you need to remove it from the car in order to use that install/removal tool. If it is loose already, you want to remove the fan belt first before removing the front plate to get at the clutch, that little bolt may be the only thing that's preventing it from being yanked off by the belt.