"No abusing the rules" probably only works if people can coordinate successfully on "the spirit of the rules".

I think one direction to explore is to have a games master picking sets that are easy to define (at least roughly), but hard to enumerate. Things like "locations in New York", "subsets of the integers", "nonempty finite subsets of the irrational numbers", "letters in non-Roman alphabets", "man-made satellites currently orbiting Earth", "models of jet plane", "movies released in the 1980s". Then teams compete to coordinate on the same sets, instead of presenting sets to each other.

You need the GM because problems can be arbitrarily complicated ("{locations in NY} X {subsets of the integers} X ..."). I'm not sure how ambiguous-membership would be handled. My first thought was that if everybody in the team agrees that something is in the set, it counts; but you need to be able to disqualify unambiguously-wrong answers, or everybody just agrees to answer "the information desk in Grand Central Station at noon" regardless of the question. I suspect you can just allow the GM to veto such answers on discretion.