Is he saying that every word has a "correct" definition?

This but with different quantifiers: some possible definitions are incorrect, as discussed in the article linked in that paragraph.

If there's a missing qualifier here then it's “in practice”. I can perfectly well randomly select an element of the power set of all things which could possibly be referred to by nouns, and declare that that is the definition of “flutzpah”, and that would be an arbitrarily defined word — which no one would ever have a reason to use to actually communicate with.