I've never looked at the wikipedia articles you mention except to look up the mass of the proton.
In my opinion, the invariant mass-energy of a system should just be called mass. For fundamental particles with no internal structure (as far as we know) like electrons and quarks, this would be the rest mass. By this definition, photons have no mass because they have no invariant mass. For a compound system, like the proton, the invariant mass is its total rest frame energy divided by c^2. The total rest frame energy of a proton is the sum of the invariant mass of the quarks times c^2, the energy of their motion in their ground state, and the energy of the virtual gluons and photons (which themselves have no invariant mass).
You say the quarks and gluons gain extra mass from their kinetic energy; I say the proton gains extra mass from the kinetic energy of the quarks and gluons. This is starting to sound like the difference between 0.999... and 1.
The only difference between your world and mine is linguistic. We're choosing to attach the word "mass" to different parts of equations. The math works out in either case.