Let $\xi=(E,p,B)$ and $\omega=(EG,\pi,BG)$ be principal $G$-bundles. Further, let $\omega$ be universal. According to Husemoller that means $[-,BG]\rightarrow k_G(-)$ is a natural isomorphy, where $k_G(X)$ are the isomorphy classes of principal $G$-bundles over $X$. If $\xi$ is universal, it is clear that $B$ and $BG$ are homotopy equivalent.
In contrast to this, tom Dieck defines universality in terms of the total spaces, i.e. $EG \rightarrow BG$ is defined as terminal object in the appropriate homotopy category of principal $G$-bundles. This means that $\omega$ is universal if for all $\xi$ there is a principal $G$-bundle map $E\rightarrow EG$, unique up to homotopy.
Does the first definition imply the second and are they equivalent?