massless or massive photon?

The usual argument for massless photons uses the Hamiltonian

H = (\sqrt (m^2c^4 + p^2c^2) ) (1)

and set (m = 0) to yield

H = pc

However, the original Hamiltonian (1) was derived using the Legendre
transformation

H = pv - L = (\sqrt (m^2c^4 + p^2c^2) )

If alternatively we start assuming (v = c) in the transformation, the
result is

H = pc - L = pc (2)

where no assumption was taken about the mass.

Is this Hamiltonian (2) representing some kind of massive photon (somewhat
as in Proca theory [#]) or is really the same that (1)?

[#] http://en.wikipedia.org/wiki/Proca_action



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