Photons in fact do have mass -- otherwise they would have zero momentum, which we know is not the case (when we bounce them into stuff they exert a pressure). In the old days, scientists used to differentiate between "rest mass" and "relativistic mass", and so they would say that a photon has zero rest mass but nonzero relativistic mass. Eventually they decided that this was not only confusing, but also unnecessary since special relativity showed us that mass and energy are equivalent.We are having a semantic argument over the meaning of "pure".
Considering energy and mass are interchangeable (and exist simultaneously), there are many possible states between pure mass at zero energy and pure energy at zero mass.
Photons are massless. Thus, pure energy.
I don't deny the mass equivalence of energy. Or the "existence" of relativistic mass. Or that a photon has momentum.
But if you accept photons propagate at c, they must be massless -- and therefore, pure energy.
Nowadays, whether a scientist calculates using mass or energy depends only on the type of work. The HEP physicists almost invariably use energy (e.g., the mass of the Higgs boson is said to be 125 GeV, which is a unit of energy), while astronomers tend to do their work in units of mass, which fits naturally with their Newtonian calculations.
The point is that mass and energy are equivalent descriptions of a particular property of a system. Neither mass nor energy are things that exist in and of themselves. It makes no more sense to speak of "pure energy" than it does to speak of "pure mass".