First of all, I don't think they are to be synonyms, since they are about different topics. It is true that those topics can sometimes overlap a bit, but at the end of the day they're different topics with vastly different questions. But I agree that at least the continuity tag is used a bit inconsistently for things where the chronology tag might also apply. This might be grounded in the less than clear and unconfusing definition of that tag.
But let's first look at the easier chronology. This seems to apply to the chronology of events in a movie/TV-show's story (to which continuity can sometimes be a key/contradiction/...). And as of now the questions with this tag seem to give a good and fitting overview of the kinds questions that this applies to.
You already give a movie-related definition of continuity, but this seems too narrow to me. I would not consider continuity only as shot-to-shot continuity in the sense of "movie-mistakes" as given by that quote of yours. In this case it would hardly apply to many questions, which would either be off-topic "hey look at that stupid error" stuff or valid questions already tagged with production-mistakes (which ought to be used only when it's clear that it's a mistake, though). I would rather see it in a broader sense of any kind of cross-scene, cross-movie or cross-episode continuity, not only in a visual sense but also story-wise, to use a variation of your own words:
discussions regarding any kind of consistency between shots, scenes, or releases (i.e. movies/episodes/seasons).
We'd still have to be careful, though, that it doesn't become an easy-out compensation for the lack of the
good old plot-inconsistency tag.
Now when looking at story-wise continuity it might be easy to confuse chronology and continuity but noone said there can't be a bit of overlap. I for myself would say that chronology is strictly about the temporal (well yeah, who guessed it) oder of events and asking what this order is or if there is one to be expected. continuity on the other hand seems to apply when there already is to be expected some kind of chronological (or otherwise) continuity in the first place. Afterall, those tags seem to be quite orthogonal and when its about temporal continuity, nobody prohibits the use of both tags.
Now let's look at the questions tagged that way. There are some where I think this definition applies well and they deserve the continuity tag. But I agree that some of them might even deserve chronology in addition to that, when they're more about the broader timeline of the events and the question if there is a continuity to be upheld in the first place:
Other questions don't have anything to do with continuity specifically and might deserve other tags:
And others seem to apply only for the chronology tag but not so much for continuity:
So all in all I don't see many problems with the current use of chronology and continuity, even if the latter might sometimes be a little misused or sometimes just both apply. But all in all, if there are situations where it isn't entirely clear, there might as well be both tags appropriate or it depends more on the way the question is asked. Not a definite answer, I agree, but it largely works currently, I think. And improving the tag wiki might not hurt.