Should the tag be split?
I think it's clear that the majority want to see popularity contests continue, but also the majority find problems with most challenges that are posted with this tag.
Is part of the problem that the tag is used for more than one thing?
There seem to be at least two different challenge types that are tagged popularity-contest.
There are restrictive conditions under which the challenge is to make the most popular answer. These are the most likely to be closed but also seem the best fit for the tag "popularity contest". Those that are not closed are among the most popular challenges on the site.
Then there are competitions which are not about being popular, but about achieving an objective, and popularity contest is chosen as the winning condition simply because whether and how well that objective has been achieved is down to human judgement. These seem a better fit for the site, but don't quite seem to fit the tag. Although the winner is still determined by votes, perhaps these should be called something else? Maybe someone can think of something better than human-judgement, which is all I can think of at present.
Pure popularity contests
Human judged contests
Here I'm splitting based on whether there is a target. Human judged contests have a specific target (either fixed or taken as an input), and the humans are necessary in order to assess how close a solution comes to that target. Pure popularity contests have no target, just a tight restriction that makes creating a solution challenging, even though it's not technically a solution to anything.
Are there better/more ways to split vote-determined contests? Are there reasons to avoid having tags to distinguish in this way?
What I have called "pure popularity contests" are what often get closed as "just an art contest". I think having two separate tags would help to emphasise that a pure popularity contest needs to have particularly restrictive conditions to drive creativity and ensure that programming skill is demanded in order to be competitive.