Photons have zero rest mass. But they're never at rest, they are always travelling at the speed of light. They have mass equivalent to their energy, which is related to their wavelength.
Mainstream academic physics has completely eliminated the concept of non-rest, variable mass. Mass and energy are equivalent, so you CAN apply mass values to the kinetic energy of a particle, but it turns out to be more confusing than its worth.
To be clear, it's just a reformulation of the math involved. Both formulations of GR/SR with and without variable mass create the same predictions. It's just that it keeps the concept of mass more coherent to stick with "mass is the energy of a particle at rest".
When the velocity is the speed of light, solving the equation of special relativity yields m of zero.
We have a successful theory of quantum mechanics which integrates with special relativity, so that definition holds, even if you take the same approach from an Effective Field Theory perspective.
The article you link starts off by defining mass as rest mass. I think the article shows very clearly that photons have no rest mass, using several different arguments.
However, a photon has energy, so therefore it must have not-at-rest mass. The article doesn't really comment on not-at-rest mass. We know photons interact with gravity because of gravitational lensing.
The calculation for mass zeros out as c is a divisor. You get zero for one of the coefficients of the mass calculation, which winds up making the whole thing zero.
The article doesn't have to comment on not-at-rest mass because the math is clear.
As for interacting with gravity, photons move through spacetime, and general relativity shows that gravity is the curvature of spacetime, and photons move on the shortest geodesic for that spacetime.
I'm not enough of a physicist to explain why this framing is wrong, but I took enough physics in college to know it absolutely is wrong.
Fun fact from my "Modern Physics" homework: a system of two photons can have mass, if they're traveling in different directions. In parallel though, they're still massless.
