Two possible processes in gravitational redshift I note there are many question about gravitational redshift and I realize that my question might be buried in one of those but I can’t see it.
I am thinking about two different processes in relation to gravitational redshift:


*

*The photon on its way out from the Sun’s gravity field is experience gravitational redshift 

*The atom is subject to gravity induced effects which slows down the atom’s processes implying that the photon is red-shifted already at emission.
Are both correct and valid ?
I read in http://www.einstein-online.info/spotlights/redshift_white_dwarfs that:
A combination of Newtonian gravity, a particle theory of light, and the weak equivalence principle (gravitating mass equals inertial mass) suffices. It is, therefore, perhaps best regarded as a test of that principle rather than as a test of general relativity.
My follow-up question is:  Are both examples of the principle of equivalence or is the second version different from the first (and perhaps in need of the full general theory of relativity for its explanation)?
 A: Both are equally derived from the gravitational redshift. In #2 the light is further redshifted if it escapes the earth. In #1, the same is true, on the Suns surface (the sun's light comes from near the surface), there is a redshift at which it is emitted, and when it escapes from the Sun it is further redshifted. The whole thing in both cases just depends on the difference in gravitational potential, that is, 
Fractional redshift = $\Delta \nu/\nu$ = - $\Delta \phi$/$c^2$
If emitted at the Sun and received at the earth take both potentials into account (from the Sun there is a semi random Doppler from the velocities of the emitting particles in the high temperature surface, much less in white dwarfs)
One point on that article you referenced. It was written by an astronomer/astrophysicist. The tricky part is the particle theory of light, which was really a Newtonian idea. Well, we know there is some truth that light can act like particles, but also that it is a wave, and the true theory is quantum electrodynamics (QED). In the particle theory of light, to have light have inertial mass you postulate that it is proportional to freq which you postulate is proportional to energy, so there is a frequency, mass and energy equivalence. The important assumption is best stated that energy and mass are equivalent because in modern terminology light has no mass, it is massless, but carries energy, thus that equivalence is needed.
Best way to derive the equations is to use General Relativity. For cosmological redshift you have to use the cosmological model of the theory. 
