A homotopy commutative diagram $\require{AMScd}$ \begin{CD} W @>\varphi_y>> Y\\ @V\varphi_xV V @VgVV\\ X @>f>> Z, \end{CD} is called a homotopy pullback, if there exists a homotopy equivalence $$\varphi: W \to X \times^h_Z Y$$ satisfying $\varphi_x = p_x \varphi$ and $\varphi_y = p_y \varphi$, as illustrated in the diagram.
My understanding is that if $Z$ is an $H$-space (in particular if it is a loop space), then such a homotopy equivalence $\varphi: W \to X \times^h_Z Y$ is always constructible, based on @Thyone https://math.stackexchange.com/a/2922633/79069.
However, my questions:
I think one advantage is that it guarantees that we also have a pullback from $X\times Y$ to $X \times^h_Z Y$ in the a fiber sequence $$ \Omega Z\to X \times^h_Z Y \to X\times Y\to Z, $$ (where $\Omega Z$ is the loop space of $Z$), based on @Thyone https://math.stackexchange.com/a/2922633/79069. Can we classify all such possible $X \times^h_Z Y$, given the data say $W \to X \times Y$? How many classes? (Related to sort of Postnikov classes?)
What else can we do other than constructing $X \times^h_Z Y \to X\times Y$?
What are the other advantage and the power of constructing this homotopy equivalence $\varphi: W \to X \times^h_Z Y$?
Note that every homeomorphism is a homotopy equivalence, but the converse is not true.