When looking at the classification of massless particles, one finds that there is the (half-integer) quantum number "helicity" $h$. For every possible $h$ there is a certain particle kind. In the case of the $h=1$ representation it is the photon which we group together with the $h=-1$ rep (because of parity invariance of the electromagnetic interaction). So is the photon we use in the Standard Model actually a set of two distinct fundamental particles?
Well, it depends how you define "distinct fundamental particle".
If you insist that Wigner's classification is what defines a particle, i.e. "particle = irreducible unitary representation of the Lorentz/Poincare group", then the photon is two particles, as you say.
But, more commonly, we do not look at the particles like this - particles arise as the states created by the modes of a free quantum field, and to every field there corresponds a particle. The photon (with either helicity) is the quantum of the electromagnetic gauge field.
Since nothing distinguishes the "two photons" except for helicity, to say that there are two distinct photons is probably more confusing than helpful.