Don’t rely on tags to provide meaning (usually).
I think the problems in these cases mainly comes from these tags being just ambiguous enough to be a problem: when one uses [rules-as-written], [system-agnostic], or multiple system tags, there's a few different things they could mean in each case. Of course, the asker and all readers could assume they've all communicated and understood the one correct meaning, but that's unlikely. In cases like the above, the situation's left ambiguous, just that nobody's asking for clarification.
In all cases, I think we need to stop treating these tags as having implicit clear meaning, because they don't. In the cases above, I'm left having to wonder what exactly the tag means there out of the possible meanings, with no clear answer available.
Whatever information the author is attempting to communicate with these tags should be in the question to make it clear and unambiguous the sort of results they want. Then the tag can be used if it correlates to what’s being described in the question. Note this isn't a suggestion to change these tags or really change the system, it's a suggestion that we treat these situations as ambiguous, and respond as we usually would: request clarification, vote to close as unclear if necessary, etc.
Someone using the rules-as-written tag should be specifying their RAW constraints. If they don't have any, they probably shouldn't be using the tag.
If someone's playing in a specific system (that they have stated in the question) and asking for help with a problem they'll be applying in that system, and they've used the [system-agnostic] tag, it should be clear from their question why that tag is there: Do they want to apply this to multiple systems? Are they using their own homebrew thing? Otherwise they should probably just be using their system's tag.
If someone's tagged their question with multiple systems or editions, they should be specifying their relationship to those and make it obvious why each tag is there.
As for our practice of not mentioning the system when we’re using just one and it’s tagged: that’s generally harmless, and I don't think we should change that. As SSD noted in comments, that will generally result in boilerplate that was not ambiguous to begin with.