If the market can remain irrational indefinitely, is it even being irrational anymore? Words should continue to have meaning.

I think the parent is referring to the saying "the market can remain irrational longer than you can remain solvent". Meaning that the market can remain irrational for arbitrary periods of time, not infinite. It'll eventually converge, but it's a question of how long.

Rationality is not defined by what the market thinks, so yes.