I think the three tags cryptic-crosswords, cryptic-clues, and crosswords refer to distinct things and each is useful. I think it's reasonable that a user might want to do an actual complete cryptic crossword puzzle, and use that tag to search for any that have been submitted. At the same time, a puzzle like this would be an odd result for a user expecting a cryptic crossword, but heavily relies on cryptic clues nonetheless.
I think it's clear that there are fans of cryptic clues around here, whether or not they're used in a standard crossword format, and so they may want to find any puzzle that has a cryptic-clue component.
I think the best way forward is your suggestion to more rigorously apply the tags. I suggested a couple changes to the tag wikis that might help and will re-tag some myself.
An added issue here is that "cryptic" is similar to "mysterious" or "enigmatic", and so the tag
cryptic-clues seems to have been applied to a lot of questions that aren't crossword-style cryptic-clues, but are "mysterious", which further confounds things...