Photons have zero rest mass.
Cite: ask any actual physicist. I hate to appeal to authority, but this is the view I was taught in grad school (PhD, UCB, 2001), and I have never spoken to anyone in physics or astrophysics who thought "relativistic mass" was helpful.
I do have a question though, if you or anyone else wants to clear this up for me. Which kinds of energy are no longer considered part of a system's "mass"? I gather that kinetic energy is not. What about various forms of binding energy?
The essential idea is that mass is Lorentz invariant, which means it does not change under changes of coordinate systems. So mass must be independent of velocity and orientation, which are just artifacts of how you measure anyways.