Yes, these kinds of gimmick puzzles are called “rebus” puzzles, because in an actual rebus, pictures are used to stand for words or parts of words. If you were solving a rebus crossword on paper, you could draw a symbol or picture in one of the squares instead of multiple letters. For example, I’ve seen rebus crosswords based on Greek letters, where you could write a symbol like \pi in a square to stand for the letters PI, or \theta to stand for the letters THETA.
Here’s a recent thread about these sorts of crosswords (which, as @Maserschmidt notes, the NYT tends to run on Thursdays and some Sundays). As you can see from this thread, some solvers love these, some hate them.