# Abstract dual of a graph and $K_{3,3}$

Let $G$ be a graph. The abstract dual of $G$ can be defined as a graph $G^*$ whose edges are in one to one correspondence with $G$ and whose spanning trees are obtained by taking the complements of the images of the spanning trees of $G$. Using this definition and Whitney's theorem (A graph is planar iff it has an abstract dual) I wish to establish that $K_{3,3}$ is non planar.

But I find another definition of dual on wikipedia according to which cycles are sent to cuts and cuts to cycles. This definition has also been implicitly used in the above linked proof for non planarity of $K_5$. So I am confused now. Given the abstract dual $G^*$ is it true that cycles of $G$ correspond to cuts of $G^*$ and cuts of $G$ to cycles of $G^*$?