Why are they called "screen" distributions?

Naturally, one should consider the quotient space $V/{\rm rad}(V)$ which consists of ${\rm rad}(V)$-rays (affine spaces parallel to ${\rm rad}(V)$). A screen space $SV$ intersects a ray in exactly one point, like a screen intersects the rays from a projector. Similarly on a manifold.

The word "screen" refers to lightlike dimensional reductions, a worldline in $(d+1)+1$ dimensional space-time is projected onto the screen $x^{d+1}=0$ and the projected $d+1$ dimensional curve is parameterized by Galilean time $t$. The projection is called the "shadow" on the "screen". For an introduction, see Classical aspects of lightlike dimensional reduction.

A particular feature of the dimensional reduction is that relativistic dynamics projects to non-relativistic physics on the screen. The history goes back to Eisenhart's Dynamical Trajectories and Geodesics (1929), who discovered the equivalence of relativistic Poincaré invariance and nonrelativistic Galilei invariance in a lightlike reduced spacetime with one dimension less.