Is left t-exactness of $Rf_\ast$ for $f : X\longrightarrow Y$ affine just Andreotti-Frenkel

Question

Sorry I'm a little lost in those perverse sheaves. I think this left t-exactness property is called Artin-Grothendieck vanishing, which I take to be the relative version of $H^n(X,\mathcal{P})=0$ for $n> 0$, affine $X$, and perverse $\mathcal{P}$. But this is just Andreotti-Frenkel basically I think, why does this need a new name? Or maybe I'm mixing up a few things.