Sure, but between "all edges have positive weights" and "there are cycles with negative total weight" there are still graphs that have negative edge weights, yet there's no negative-weight cycle.

Example:

V = { A, B, C }

E = { A -> B, B -> C, C -> A }

weight(A -> B) = 1

weight(B -> C) = 1

weight(C -> A) = -1