I don't know enough to give a proper answer, but hopefully someone will correct me if I'm wrong. I think the issue is: does the fact that light, in some sense, travels slower in some medium than in a vacuum mean that (1)
c is reduced in that medium to the speed of light in that medium, or (2) does light travel slower than
c in that medium while
c is the same everywhere and at all times? And I think the answer is (2).
There's a maximum instantaneous speed, a cosmic speed limit,
c, which is a physical constant. Anything going faster than this limit could violate causality; it would create a situation where two events could be causally related, but there'd be no natural way to decide what order these events happened in. If you analysed the interaction in one reference frame and decided that event A caused event B, you could transform coordinates by a continuous Lorentz boost to another reference frame in which event B happened first. But there are mediums in which particles can travel faster than light can through them, and as far as I know this isn't considered paradoxical, so presumably it's the light that travels slower than
c, rather than
c itself being reduced. See http://www.phys.ncku.edu.tw/mirrors/physicsfaq/ (Is the speed of light constant? + Is it possible to go faster than light?)
Another FAQ entry of relevance, although it doesn't directly address this question:
https://www.physicsforums.com/showthread.php?t=104715 (Do Photons Move Slower in a Solid Medium?)
I'm afraid I don't understand whether the above link is broadly agreeing with the idea that photons are absorbed then emitted after a delay, or not. It says they aren't absorbed "by the atom via an atomic transition". On the other hand, "when a photon encounters a solid, [...], this photon can be absorbed by the solid". So is this photon emitted after a delay? It seems like the obvious question, but they don't say. On the other hand, a photon that is
not absorbed by the lattice "is re-emitted but with a very slight delay." I had to read that a couple of times. Typo? I don't know. Probably just my lack of the background knowledge needed to make sense of it. They go on: "This, naively, is the origin of the apparent slowdown of the light speed in the material. The emitted photon may encounter other lattice ions as it makes its way through the material and this accumulate the delay."
So they do call the slowdown "apparent", and do talk about re-emission, and do talk about delays associated with re-emission of photons, and do talk about absorption of photons. Is any photon emitted the same photon as one that was absorbed? I wondered if they might be making a philosophical point about that, and saying no, it's a new photon. But that might clash with the word "re-emission", so I just don't know...
