Considering what is taught in the Academy now and in annual refresher training, the new wave is slide stop - the thinking is to go over the top with the weak hand to release the slide on the fresh magazine. Personally I'm old school, and prefer a slide release I can work one thumbed. The P-10C slide stop seems deliberately designed to not be used as a slide release. It's very thin and the top of it is not angled/beveled/plateaued to allow for the downward motion of a thumb, except in dire circumstances. If you hit the slide stop during range here, the instructors get all unglued. Considering I started shooting before many of them are bored, (and I outrank most of them), I smile and keep moving on. The theory was explained to me that continually forcing the slide stop down can eventually round out the notch on the slide so it won't hold it any more. They could not give me an estimate on how long it would take to do something like that, of course. I would like to see a slide release built instead of the slide stop - did you note in Cajun Gun Works subforum that they are looking into that thing right now?
I have heard it called a magazine catch before, but the speaker was much older than me, so that term might originate way back in the day. I agree that when I was learning to shoot the two controls were always known as slide release and magazine release.