Electron density definition of single, double, triple bonds?
Below are plots of the electron density for staggered ethane, ethene, and ethyne, computed from electronic structure theory.  For ethene, which is planar, imagine that the hydogens are jutting forward and backward into the paper, so that the contour plots should contain any putative $\pi$ bonds. It is the orthogonal plane that still contains the two carbon atoms.

A couple of points that I wish to make:


You are correct (in this case for sure) that there is some association between stronger bonds and higher electron density at the C-C bond critical point $(\star)$.  This was noted by Bader and co-workers.
There is no evidence (to my eyes) that there are any side-wise $p$ orbital interactions forming $\pi$ bonds. These look like single $\sigma$ bonds!


My question is, where do we draw the dividing line in the correlation between electron density at the bond critical point, and what constitutes single, double, and triple bonds?



Responding to @michaelm: I might have expected this, if for example, ethene were to have a "double bond" between the carbons: