All online IQ tests are fake, no exception. Most "Mensa online tests" are unrelated to Mensa; they just say so, because it makes people more likely to click on them.

A serious IQ test would look like this: https://en.wikipedia.org/wiki/Raven%27s_Progressive_Matrices

> I can't fathom it being anything other than an indicator on how well someone can solve these matrix association problems.

Well, this is one of the interesting findings about intelligence. Turns out that how well people can solve these tests, correlates strongly with many other things.

At some moment in history, there was a competing theory of "multiple intelligences", assuming that the kind of intelligence for doing math is independent from the kind of intelligence for solving verbal problems. But when people were given different tests, it turned out that the people who score high on one kind of test are also likely to score high on the other kind of test.

So the reason why the "matrices" tests are so popular is that it doesn't matter much what kind of test you use, and the matrices are less culturally sensitive (i.e. in English verbal test, it would matter a lot whether English is your first or second language).

> what would be a rigid definition of IQ?

Approximately: That thing that allows you to do better in a wide range of problems. (As opposed to a talent or learning disability, which is that thing that allows you to do better/worse in one specific thing.)