Usually "non-Euclidean" is used for Riemannian manifolds, so if something is not a Riemannian manifold, it is not called Euclidean or non-Euclidean. Just like

Usually "non-Euclidean" is used for Riemannian manifolds, so if something is not a Riemannian manifold, it is not called Euclidean or non-Euclidean. Just like "irrational number", which is not any number that is not rational -- we still assume that it is a real number, so i or Aleph-Null are not considered irrational. If we include teleport points in our metric, we no longer have a manifold (if teleports are one-way, not even a metric space). Similarly for Mountain Valley, or affine/projective manifolds that can be found in some games, it is wrong to call them Euclidean or non-Euclidean. Anyway, originally "non-Euclidean" was just the hyperbolic plane (the only geometric structure which satisfies all the Euclid's axioms except the fifth), and then (depending on whom you ask) it was extended to also mean other situations which are different but similar in nature -- so spherical geometry (which is just the opposite), Nil/Solv geometry (which also play with parallel lines), but not cylinders, affine manifolds, teleports, taxicab metric, and so on (they don't really do anything interesting to parallel lines).