Those are the three distinctions we have for cardinalities. For most people uncountable would usually mean $2^{\aleph_0}$. But even for set theorists, given a model of ZFC, the finite sets are finite, the countable sets are countable, and the rest is madness$^*$.
Replacing cardinality by topological-measure theoretic properties of subsets of an ordinal $\kappa$, there are non-stationary sets (small); stationary sets (big, but not too big); and clubs (which is practically everything).
$^*$ Good madness! Like a smile without a cat.