A summary of the basic development in the OP cover the points:
§ 1. A Hypothesis as to the Physical Nature of the Gravitational Field
§ 2. On the Gravitation of Energy
§ 3. Time and the Velocity of Light in the Gravitational Field
of the Einstein's 1911 paper "On the Influence of Gravitation on the Propagation of Light", just
by using Newton's theory of gravity, E =mc² and E = h.f (the first 5 pages of such paper).
SUMMARY OF EQUATIONS IN THE OP AT THIS THREAD:
Using Newton's Equivalence Principle, as it was applied since 1700s, we have:
d/dt(Mi.v) = K . Me.Mg/r² , as Mi = Mg is invariant with time, then
Mi.dv/dt = K . Me.Mg/r²
g = d²x/dt² = K . Me/r²
Being the Potential Energy for a mass M at an height h:
U = F.h = M.g.h = K . M/h² (for low values of h),
and being the energy of a mass M at ground level equal to E1 (an arbitrary value before 1905),
then, the total energy E2 at an height h, being M stationary is:
E2 = E1 + U = E1 + M.g.h
Einstein, on his 1911 paper, made use of E = mc² appeared by 1911 and, using Newton Equivalence Principle
stated:
"Given the certainty that M = E/c² and that Mi = Mg, I declare that the
following assertion is true:
Given the energy E1 of a mass M at ground level being E1 = M.c², obtained:
E2 = E1 + U = E1 + M.g.h = E1 + E1.g.h/c² = E1. (1 + g.h/c²) , the equation (1) at § 2.
as g.h = K.Me/(R+h) = Φ (the newtonian gravitational potential), then the equation (1a) is obtained:
(1a) E2 = E1. (1 + Φ/c²)
and, using Planck's E = h.f, the equation (2a) at § 3 is obtained:
(2a) f2 = f1. (1 + Φ/c²)
Now, extending the reach of the OP, in the final part of § 3 Einstein ventured this hypothesis:
"If we call the velocity of light at the origin of co-ordinates c0, then the velocity of light c at a location with
the gravitation potential Φ will be given by the relation
(3) c = cₒ. (1 + Φ/c²)
The principle of the constancy of the velocity of light holds good according to this theory in a different
form from that which usually underlies the ordinary theory of relativity."
NOTE: I used the same name for this equations and the ones that appear on the 1911 paper. It's important
to observe that (3) is derived from (2a) by just multiplying both sides with an arbitrary wavelength λ, so that
c = λ.f2 and cₒ = λ.f1 (value of c at ground level)
This final concept, from 1911, didn't gain any traction but the concept of equation (2a) persists until today.
The final point "§ 4. Bending of Light-Rays in the Gravitational Field" gives a deflection α = 0.75" (same value
and equation than that of von Soldner, 1801):
α = 2k.M/(Δ.c²)
which is exactly HALF the value of the 1915 equation, which gives α = 1.5" (allegedly measured by Eddington in 1919)
α = 2Φ/c² (with Φ as the gravitational potential in the surface of the Sun).
which is TWICE the value obtained in equation (3), by obtaining the relative difference of the speed of light under Φ :
(c - cₒ)/cₒ = Φ/c² = α/2
Just saying. Einstein liked to play with elementary algebra. Maybe, a "passing by" effect on light path lit his eyes by then,
thinking how to conserve the constancy of the speed of light in his prediction of deflection by the Sun.
At point § 4, he wrote:
"The angle of deflection per unit of path of the light-ray is thus
- 1/c . ∂c/∂n' , or by (3) - 1/c² . ∂Φ/∂n'
Finally, we obtain for the deflection α; which a light-ray experiences toward the side n' on any path (s) the expression
α = - 1/c² . ∫ ∂Φ/∂n' ds = 1/c² . ∫ k.M/r² . cos θ ds = 2k.M/(Δ.c²) [ ∫ between -π/2 and +π/2 ]
where k denotes the constant of gravitation, M the mass of the heavenly body, Δ the distance of the ray from the center
of the body (and r and θ are as shown in Fig. 3). A light-ray going past the Sun would accordingly undergo deflection by
the amount of 4 .10^6 = 0.83 seconds of arc."
NOTE: ∫ cos θ/r² . ds = 2/Δ [ ∫ between -π/2 and +π/2 ] is a fudged result. Anyone who may check with Fig.3 in the paper,
will observe that r.sin θ = S, so (ds = dr . sin θ + r . cos θ) and the integration of an UNKNOWN path between -π/2 and +π/2
gives an ARBITRARY value, which was fixed as 2/Δ to equate von Soldner value. Probably Einstein thought that nobody was
going to claim for additional explanation upon this final part.