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][1] 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 thread][2]. 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.'


  [1]: http://arxiv.org/abs/1305.0086
  [2]: http://mathoverflow.net/questions/85329/hoveys-unit-axiom-in-monoidal-model-categories/85995#85995