In the book Fibre Bundles by Husemoller, universal G-bundles are introduced as bundles over a homotopy type $BG$, for which the cofunctor $[-,BG]\rightarrow k_G(-)$ is a natural isomorphism.

Contrary to this, tom Dieck defines universal G-bundles as those for which the total space EG is terminal for some homotopy category, i.e. for all numberable free G-spaces $E$ there is a G-map $E\rightarrow EG$, unique up to G-homotopy.

How does the first definition imply the second? My aim is to get a unique homotopy type to show that any total space of a universal bundle as defined in the first way is contractible without restricting to CW complexes.

I have already unsuccessfully asked this question here https://math.stackexchange.com/questions/2416706/total-spaces-of-universal-principal-g-bundles