I would call them all cheese graters. Plus the kind my mother brought home once that looked kind of like a spaceship. It had three legs that folded down so you could put it right over the bowl you were using and a kind of a hatch to put the cheese in. I tried to find a picture on teh interwebs, but I don't think that particular model was successful.
Mouli (Moulinex?) was the dominant (or perhaps only) brand to make this particular style of grater, with the rotating cylinder, the cavity for the food and the pressing arm. Could be used for cheese, but more commonly used for other foods. Not so popular anymore, except in Europe.
Whereas in my neck of the woods the "cheese grater" was and remains a simpler thing, with no moving parts. More like these.