That terminology is outdated. Today, mass generally means invariant mass, corresponding to rest mass in case of 'massive' particles. The invariant mass of photons is 0, hence it's 'massless'.
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.
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