I dunno if I can answer properly but the answer may lie somewhere here:
The speed of light is the same for all observers. It's independent of the motion of the light source and of the frame of reference of the observer. So maybe "which time did he use" is irrelevant.
How can the speed of light be a constant (absolute) if the time is relative?
Time is inseparable part of spacetime and space involves mass/gravitation. LIght has no mass. The speed of light constant is more generally the speed of all mass-less particles, which includes light.
Maybe the answers to your questions are in there somewhere; thanks for the question.
Actually, photons do have mass, although some "purists" insist they only have a momentum, or as we used to say in the Navy "all vector, no force!" :). But we know they have mass because photons are affected by gravitational pull (see "light bending," as well as black holes).
But here is the problem: we know that light doesn't travel at the same speed in different media, such as glass. If it didn't lenses would be worthless.
Snells law or refraction is validated by the fact that light does slow down when it encounters glass. Such encounter causes the wavefront to become "deformed" or "retarded" relative to the portion of the wavefront that is still traveling outside of glass.
This "retardation" or deformation of the wavefront is responsible for causing a plane wave to become spherical with focus at its center. Also, depending on the shape of the surface encountered, it can become aspherical and cause optical aberration at the focus.
A skillful optical engineer will introduce another element, glass or mirror, or even set of elements, into the light path to deform the deformed parts of the wavefront so that that it returns to a spherical front,and form an aberration-free image of an object.