Since I got no responses to this question at Stack Exchange, please let me try my luck here.

Call a continuous map $\pi:E\to B$ between CW complexes a homotopy fiber bundle if for any $x$ in the image of $\pi$, there is an open neighbourhood $U\subset B$ of $\pi(x)$ and homotopy equivalence $\pi^{-1}(U)→U\times F$ over $U$.

I don't know if this has a different name in the literature or even if it is reasonable. Replacing ''homotopy equivalence'' by ''homeomorphism'' should be the definition of an ordinary fiber bundle.

How relates a ''homotopy fiber bundle'' to the notion of a Serre fibration?

At least both properties imply that the fibers over connected components are all weakly homotopy equivalent.