You're using outdated terminology. Modern physicists use "mass" to refer to what was called "rest mass" shortly after relativity was developed in order to distinguish it from the new concepts they at the time called "relativistic mass" and "inertial mass". The term "relativistic mass" was motivated in part by the desire to still think of all gravitation as being sourced by some sort of mass, but the modern terminology is to say that gravity is sourced by the stress-energy tensor, of which (rest) mass is just one component.
Modern physicists unequivocally say "a photon has zero mass, but has non-zero momentum and energy".
I have a physics degree and still say that photons have zero rest mass, and then parenthetically mention that they never rest, precisely because of the issues of communication that tend to arise when we talk about things like thrust.
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.
It is not false, because I made clear what mass I am talking about. Dogmatic nit-picking may satisfy your wish to feel superior to the great unwashed masses, but it is not particularly helpful.
I am aware of Okun's campaign to eliminate the term "realtivistic mass". I am not sure I am 100% on board with this, because the concept of "relativistic mass" is intuitive and practically useful (used by engineers working on particle accelerators, for ex.).
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).
Modern physicists unequivocally say "a photon has zero mass, but has non-zero momentum and energy".