Good question; one of the items on my TODO list was to organize these in a better way, together with the related tags study (where it concerns endgame studies) and studies.
In my opinion, we should follow the Wikipedia definitions. This article lists a quite exhaustive type of chess problems, of which the most common are:
- Directmates (straightforward mate in n)
That list contains endgame studies, but they are (in my opinion) sufficiently different enough to warrant their own tag. Note that the tag excerpt for puzzles is actually something that I would definitely call a 'problem', and Wikipedia agrees with me:
Problems can be contrasted with tactical puzzles often found in chess columns or magazines in which the task is to find the best move or sequence of moves (usually leading to mate or gain of material) from a given position. Such puzzles are often taken from actual games, or at least have positions which look as if they could have arisen during a game, and are used for instructional purposes. Most such puzzles fail to exhibit the above features.
Puzzles would be a good for the (often) tactical middlegame puzzles 'White to play and win', where the solution is often a line winning a piece or so. On the other hand, it's also a good determiner for chess-themed puzzles like this one. I haven't found a good solution for this
problem dilemma yet.