Modern physicists unequivocally say "a photon has zero mass, but has non-zero momentum and energy".
I'm always tailoring it to the audience because sometimes physics metaphors need to be unpacked when communicating. I have run into people who think that the "hole" that can move around in a lattice is an actual particle, not a quasiparticle, like it was a swimmy little positron just sitting in there, simply because the metaphoric shorthand had overtaken reality.
If I were talking physics grad to physics grad, I wouldn't do that. Anyone else? Yeah, brushing the complexities under the rug with shorthand leads to, well ... the idea that you can get thrust without kicking something in the other direction.
For instance, if a quantity of photons is put into reflective mirror box, when you put it into the scale, the measured weight will increase by E/c^2 (both before and after the photons are absorbed).
The gamma factors hidden by the relativistic mass definition of course still occur, but they get folded into the 4-vector quantities instead, which happens naturally if you use proper time for your derivatives.
This way, we avoid having to introduce things like transverse and longitudinal mass, which was the explanation within the variable-mass picture as to why the spatial part of 4-acceleration generally isn't parallel to 3-acceleration.
edit: s/velocity/acceleration/ where appropriate