Sorry, this is not on-topic here, so I'll keep my asnwer brief (for me)
I learnt my relativity from popular books
Oh dear
But Baez's site is very sound.
First, both formulations "mass equals rest mass" and "mass equals relativistic mass" are formally equivalent. The decision as to which to use is a matter of convenience rather than of fundamental disagreement.
No, absolutely not. Rest mass is an invariant quantity. Relativistic mass is an observer dependent quantity, and hence largely meaningless. If I ask you for your height, what do I expect? A measurement depending on your particular position? On how far away you are from me? On how fast you happen to be moving at that time?
I don't see why we should not use this example to justify the claim that light has mass in general
Because mass is not the only generator of inertia and gravitation. In this case one could say that it is the momentum of the photons that is causing the difference - it is certainly not their mass as that is zero. The trouble is there are too many uses of the word mass being floated around. The "mass" of a composite body is the gravitational "mass" - something that we actually often call stress/energy to avoid confusion. This is very different to the rest mass of a particle and to the relativistic "mass" of a particle.
None-the-less, you are wrong about the definitional convention on mass used by Meyer in his physicality implies mass criterion. Using the convention Meyer must be using if his criterion is to be even coherent, photons do have mass.
Meyer is simply talking nonsense and he has no clue to the real nature of mass and its various defintions - it's not his field. Why would I be intersted in any of his definitions?