When I created the first puzzle-suffix tag during the private beta (calculation-puzzle), it was meant (at least in my mind, as other people created more of them as well) as part of a general classification system for all puzzle questions on the site, similar to the way all questions on the meta must be tagged with at least one of bug, discussion, etc. The intention was that (almost) every puzzle would fall into one of these categories, any the tags without puzzle suffixes would denote features of a puzzle, such as involving graphics or outside information.
It's obvious that this isn't actually happening here. Specifically, I'm thinking of the case of calculation-puzzle itself, which has outlived its usefulness and should be renamed to arithmetic, but should still be kept separate from mathematics as I mentioned in this question just now.
But while I'm at it, here's a list of all the puzzle-suffix tags that are currently in use on the site for actual puzzle questions, in order of usage:
These tags don't quite aim to cover all or even almost all categories of puzzles, and some of them (everything after "computer-puzzle") aren't even broad enough categories to deserve the -puzzle suffix.
I think we need some reorganization around this — either to remove the puzzle suffix altogether, or to redefine it well enough using some of the other, currently non-suffixed tags so that it can serve as a useful classification system.