Also, you're completely right that k-vectors don't encode shape information and that you can deform the parallelograms representing them arbitrarily. I glossed over that in the post, but now that I'm rereading it, it really sounds like too much simplification. Also, I kinda ignored the non-blade-k-vector thing for simplicity (there's the nice example of the spacetime bivector \(\gamma_0 \wedge \gamma_1 + \gamma_2 \wedge \gamma_3\) that can't be represented as a parallelogram).
And I also fully agree with your intuition for k-forms! What I've started doing is imagining 2-forms as a kind of "extruded raster" with a specific orientation, which you can intersect with a bivector, and then you look through it and count the "cells" that are "blocked" by the bivector (I've tried to depict what I mean in Blender). But that was sorta out of the scope for the post :D
