Simple discussion of photon mass
It's one thing to discuss the possibility that a photon has zero rest mass, but the level of that discussion entirely depends on the participants. Any discussion of "proof" or "definition of mass" is either going to be incredibly advanced, or little more than idle speculation based on ignorance.

I'm not seeing the "advanced" part, here.

Any discussion of photons and rest mass should at least bring up the standard special relativity equation relating energy and momentum:

This basically means that photons have momentum. If you bombard an object with photons, it will push that object (very lightly). This is also what comprises radiation pressure (which is not the same thing as the solar wind, which is a plasma physics phenomenon).

If a photon has "zero rest mass", that means that all of its energy is directly proportional to its momentum. "Proving" that the rest mass is zero is necessarily tricky: how does one design an experiment to conclusively demonstrate that the mass is zero? One can always chime in, "Well, you only determined that it's close to zero - it might be closer to zero than you can measure." This kind of thinking makes the concept of zero rest mass essentially not provable. And you can't use concepts like, "Does it exert a gravitational force?" because in general relativity, photons

*do* exert gravitational force, due to their momentum (it's part of the mass-energy density). Karl Popper addressed these concerns long ago: we can disprove whether a theory is true, but we cannot prove conclusively that it is true. Therefore, the job of science is to come up with disprovable hypotheses and try to disprove them. Those hypotheses that continued experiments fail to disprove are considered to be "true". It is always possible that any given theory might be disproved later, thus to bring this possibility up in any sort of scientific discussion contributes absolutely nothing.

A more clear indication that the rest mass of a photon is very likely zero is that the formulation of quantum electrodynamics (QED) breaks down if it's nonzero. QED has accurately predicted the behavior of electrons and photons to one part in a billion, which is rather extraordinary, considering that quantum mechanics is so "fuzzy", so if one is to postulate and get others to accept the notion of nonzero rest mass for a photon, one will have to reformulate QED and explain why it's OK that a photon has mass.

These are nontrivial tasks. Given that the current evidence indicates that the photon's rest mass is zero, the burden of proof lies on those who would postulate otherwise, just as those recent experiments showing neutrinos traveling faster than the speed of light had to bear the burden of proof for violating special relativity. One cannot just say, "It might be zero," and have any meaningful discussion, without pursuing at least some of the implications that derive from its being nonzero. And even then, the only real proof that it is nonzero would be derived from experiment, not argumentation.