Lots of good info.
However, how would you tell a butter dish without a strainer from a covered cheese dish? The ones shaped like a wedge of cheese are a no-brainer, but what about something like a brie that is in a round shape? You wouldn't necessarily have put it in a wedge dish (at least not if you had money!)
Maybe the cheese dish is more like a plate with a cover on it, rather than having the ridge and depression like a butter dish? Cheese doesn't need to be kept cold, is usually served at room temperature so there is no need for the ice to keep it cool.
Now that I think about it, maybe the covered dish I have with my breakfast set is a butter dish without the strainer? Still think it is the covered plate because the rest of the set is individual-sized, but maybe not? Have added photos, and one of the cup and tea pot for size.