It is a misconception that things are ill-defined just because they are at a quantum scale. The proton has an electric charge distribution. The definition of the proton size used here is the root mean square charge radius of that charge distribution.
Indeed, but it's actually a little bit hard to define what that charge distribution is (or in which frame...). The way it shows up, both in the cross section for elastic lepton scattering as well as in spectroscopy, is via the related quantity "form factor", as the slope at zero four-momentum transfer. While the form factor can be thought as the Fourier transform of the charge distribution for heavy objects (say, iron nuclei), for the proton, this becomes dicey.
