Maps $f$ such that every pushout along $f$ is a weak equivalence were called couniversal weak equivalences in the preprint Homotopy theory for algebras over polynomial monads by Michael Batanin and Clemens Berger. In left proper model categories such maps are characterized in Lemmas 1.5 and 1.6.
I'm not sure I agree with calling these maps 'flat' as that word is already so over-used. For example, this terminology could easily cause confusion in examples like $Ch(R)$ or the stable module category. Furthermore, a common axiom for monoidal model categories is that whenever $X$ is cofibrant and $f$ is a weak equivalence, $X\otimes f$ is a weak equivalence. Motivated by the examples above, this axiom has sometimes been called the axiom that 'cofibrant objects are flat.' I discussed this axiom a bit at this mathoverflow threadthis mathoverflow thread. I don't know if that terminology will stick (in the paper above this axiom is called the Resolution Axiom, and in a preprint of Pavlov and Scholback it's called the left cow axiom), but it's more evidence to avoid using the already-saturated 'flat.'
