Does this definition resolve the problem posed by the OP, that competence in one of various different specific activities requiring strength doesn't imply competence in the others? That is, after all, the basis on which IQ tests are attacked - competence on Raven's progressive matrices doesn't imply competence at the Piano.

"imply" is a word that suggests you think about whether it makes sense that there a causal relation between the two task. That's not central for IQ. g is a statistical construct that does things that aren't obvious.