I am not sure how to use this problem-solving tag, since by name it seems to me that any problem/excercise would fall under this category?!
I think the "problem solving" tag should be for questions about problem solving in general. For example, "what are the best strategies for solving inequalities?". It should not be used for concrete questions that simply use problem solving techniques, if those concrete questions are not about problem solving techniques. So I would say "problem solving" tag is for meta discussion about problem solving itself, not for questions that merely want to apply problem solving techniques. Otherwise, as the question here says, every single question could be tagged as "problem solving".
I think similar things about the "proof strategy" and "proof writing" tags. These should not be used on questions solely because the answer is a proof - they should be reserved for questions about proof strategies and proof writing.
The answer appears to be:
this is a meta-tag that indicates contest-type problems that are normally associated with AoPS. this is a tag for the questions about problem-solving strategies in general.
The AoPS style questions can be marked with "contest-math". It's not clear to me whether contest-math should be used as a standalone tag, or in conjunction with subject tags.
Sub-question 1: should the tag be applied to questions like this one when the OP does not do it? Such problems have some distinguishing features (i.e., nothing is expected in terms of "motivation"), so having a clear label could be helpful to voters.
Sub-question 2: is the set of AoPS-suitable problems a subset of the set of MSE-suitable problems? My guess is that the answer is yes.