Laplacian of the electron density
Yes, and no, and maybe.

Oftentimes the sign of the Laplacian at the bond critical point in terms of the quantum theory of atoms in molecules (QTAIM) is taken as a measurement for the covalency of a bond. That it is not that simple is quite logical from your question alone.

Just as in bonding in general, i.e. there is a spectrum from pure covalent (ideally, but not achievable) and pure ionic (ditto),* there cannot be a clear yes or no answer here, too. I think the case of $\ce{BCl3}$ versus $\ce{AlCl3}$ and respective dimers, which I have written about, demonstrates this feature and how QTAIM can help understand bonding. As with many things it certainly is hard to draw a line. In that sense it is probably a good criterion to judge the nature of a bond. Judging only based on the sign of the Laplacian, while being a rigorous and unambiguous way to look at things, might lead to the wrong conclusions though. 

What does it matter anyway? In many cases such a strict separation is not helpful. It should be clear, that there are covalent and ionic contributions in every bond, and reactivity often does not care about that. You'll never obtain a complete picture, if you are looking at a single property.

* And then there is charge-shift bonding (What is charge shift bonding?), but I personally don't see any value in that.