I think if you look at "dependent" as the negation of "independent" instead of the other way around, it'll make sense to you.
Independent is the lack of any dependence. So if there is even the tiniest dependence (between only a subset of the vectors), the whole set of vectors is dependent.
Your proposed definition requires a dependence relation between all of the vectors, which is just a "higher level of dependence" (so to speak) than is required to negate "independent".
