The section that you clipped on non-zero rest mass photons isn't really pertinent to the discussion. John Baez is talking about something entirely different.
Once you get past Sophmore Physics, you always mean 'rest mass' when talking about mass, unless you are in a specifically non-relativistic context. (Where it doesn't matter which one you use.) Physicists just call it 'mass', since they can all do Special Relativity in their sleep and they get tired of saying 'rest' all the time. People only talk about mass increasing as you go faster as a way of introducing Special Relativity to students. After that, it's all 'mass' means 'rest mass' from there on out.
Once you get past Sophmore Physics, you always mean 'rest mass' when talking about mass, unless you are in a specifically non-relativistic context. (Where it doesn't matter which one you use.) Physicists just call it 'mass', since they can all do Special Relativity in their sleep and they get tired of saying 'rest' all the time. People only talk about mass increasing as you go faster as a way of introducing Special Relativity to students. After that, it's all 'mass' means 'rest mass' from there on out. Sorry. To me, mass and rest mass need to be distinguished, but that just me. I thought Baez was discussing the rest mass of a photon. Maybe I'm mistaken.