Yeah, usually "negative" qualifiers in mathematics mean some specific things. "Irrational number" does not mean numbers like i or Aleph-Zero even though they are not rational numbers. It means a *real* number that is not rational. Discovery of such numbers was quite important in math. Same as with non-Euclidean geometry, the discovery that all the axioms could be satisfied but not the fifth one was important again. So it originally meant just that, not satisfy the 5th but satisfy all the others. Some people extend this but they usually try to keep the spirit (hard to explain Nil geometry using axioms, but the spirit of parallel axiom is no longer there...). Sometimes "non-X Y" is meant to include "X Y" as a special case :)
Anyway, thanks for the nice game! I have added a mention to my article.