Gravitational light bending

I’m trying to follow the next proof referred to the bending of light in a gravitational field (calculated using exclusively the Einstein’s equivalence principle).

Suppose you have an observer in an elevator (A) in free fall in a gravitational field. As this observer is in an inertial local frame, if he were to "shoot" a photon to the other side of the elevator, it would travel in a straight line. Now, imagine another observer is watching these events from a rest position in the gravitational field, outside of the elevator (B). As (A) is free-falling, he (B) regards the photon as falling with the elevator and hence, it describes a curved trajectory.

Now, as the "ticks" of a clock, or equivalently the frequency of the light in a gravitational field come modified by:

$$Delta t=(-g_{00}(x))^{frac{1}{2}}dt Rightarrow frac{1}{dt}=(-g_{00}(x))^{frac{1}{2}}frac{1}{Delta t}=(1+2phi)^{frac{1}{2}}frac{1}{Delta t},$$

where $phi=frac{GM}{r}$ is the gravitational potential. For weak gravitational fields we have $sqrt{1+2phi}approx 1+phi$ (via a Taylor expansion) and hence:

$$frac{1}{dt}=(1+phi)frac{1}{Delta t}.$$

From this point, my teacher straightforward leaps to:

$$c=c_0left(1+frac{phi}{c^2}right).$$

With the only hint that, as the $x$ coordinate is orthogonal to the gravitational gradient, it’s possible to do $dx=Delta x$.

My best guess is that $Delta t$ and $Delta x$ stand for the time (proper time) and distance in the inertial local frame, so that, multiplying by $dx$ we can get:

$$frac{dx}{dt}=(1+phi)frac{Delta x}{Delta t}.$$

So that the derivatives would be the speed of the photon, and the $c^2$ dividing the potential it’s explained since we were working in $c=1$ units and then he reverted the change, leading to the desired equation:

$$c=c_0left(1+frac{phi}{c^2}right),$$

where $c=frac{dx}{dt}$ and $c_0=frac{Delta x}{Delta t}$. What I don’t really understand is how can I explain these two last identifications. What does $c_0$ reppresent? Is it $c$ still the constant value of the speed of light in vacuum? Does this mean that the light travels faster or slower in the inertial local frame? And how does this connect to the gravitational bending of light (that’s the thing were supposed to be studying)?