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 fo

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.