I agree with your conclusion, but IQ, maybe is just a simplistic definition. IQ is more like a persons ability to conceptualize topics. It will flow throughout, if their focus is math they will have an easier time conceiving the abstract math equations, for an engineer it might be an easier time visualizing a 2d drawing as a 3d model...

But, like you noted, without the other forms of intelligence, a person might be able to conceive these things but without an ability to express them or to work with others, that skill gets lost.