I was wondering: in the languages you guys know, does "potato" refer to only the common potato or to other things, like taro or sweet potato (yam), for example? In Japanese it would make a difference, because jagaimo is the common potato, but imo is a more generic term for various types.

It refers only to the common potato. There are different names for the other types. We could define yam and tuber as additional words. My dictionary defines tuber as "a swelling on the stem or root of a plant, in which food is stored: Potatoes are the tubers of the potato plant."