They state that only as a disclaimer, after a bunch of other statements that hint at the opposite. Just adding a disclaimer to misleading writing merely makes it confusing and unclear for those who are paying attention, and still misleading for those who don't notice the disclaimer.

I think the writer might have been meaning to imply that there's a spectrum of properties from countably infinite to dense to uncountable, where each property is stronger than the previous one. But this is actually not the case. The Cantor set is uncountable, but nowhere dense. So density is not just something that's in between countability and uncountability---it's an orthogonal thing.