>Intelligence is also not skill, but the thing that makes you skillful in all cognitive tasks.

Careful with that "all", even the most highly intelligent humans still have peaks and deficits in different domains.

It's a matter of the definition. The general factor of intelligence, which is measured through various somewhat lossy proxies like IQ tests, is exactly the degree to which someone exceeds expectation on all cognitive tasks (or vice versa).

The interesting finding is that this universal correlation is strong, real, and durable. Of course people in general have cognitive domains where they function better or worse than their g factor indicates, and that's before we get into the fact that intellectual task performance is strongly predicated on knowledge and practice, which is difficult to control for outside of tests designed (successfully, I must add) to do so.