Photons have no rest mass, but one of the key points in GR is that mass and energy are interchangeable by E=mc^2. Mass-energy equivalence is how Fission bombs work.
"Relativistic mass" is one of those concepts that was considered, bandied about in the 70s for a while, then ultimately discarded as not helpful.
Photons have no mass, period. An ensemble of photons may have a non-zero invariant mass, but there's just no way in which it's correct to say a photon has a mass.
This has nothing to do with saying that you can't convert mass to energy. Electron-positron pair creation from ultra high energy gamma rays has been a known thing for a long time. Pulsars are thought to run partly off of pair creation off the magnetic field, etc.
Stupid question maybe, but if you can create mass from photons, wouldn't that start to affect the shape of the universe if it happened enough? Could you use this, in theory, to turn an ever expanding universe into one that contracts? Just keep turning energy into mass?
Nope. In GR, energy has as much "gravity" as the equivalent mass, so it won't affect anything. (Energy is more tied to the expanding universe, though, as the wavelength increases too, so it's an important distinction in early-universe cosmology.)
Rest mass is mass. Photons have zero mass. "Relativistic mass" is more confusing than helpful and is generally not used in modern GR.
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'm entirely on your side on this (PhD, UChicago, 2006), but there are still a bunch of physics teachers out there (mostly not doing research in related fields, is my rough impression) who will vigorously defend teaching the "relativistic mass" concept. They raise some decent points, but in my opinion none of it comes close to outweighing the arguments in favor of treating mass as an invariant property.
If this is a recent terminology change, then people who use the old terminology are not factually "wrong", they just use an outdated definition.
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?
Only kinetic energy does not contribute to mass. In fact, it's even narrower: only the kinetic energy of the center of mass does not contribute. For example, heat energy is really just kinetic energy of the constituent particles, and heat definitely contributes to mass.
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.
