There is a good description of what is going on in this Stack Exchange post:
http://physics.stackexchange.com/questions/13738/propagation...
It explains why saying "c is the speed of light" makes sense, because when we say light is traveling more "slowly" through a material, we are including the time spent interacting with the material, being absorbed and re-emitted.
I'm bristling a little at your statement that "the speed of light is not constant". Imagine two men walking at the same speed from A to B. But one of them is walking in a straight line, while the other is zig zagging. It would be fair to say that the one walking in a straight line is travelling from A to B faster, even though they are both moving at the same speed. The speed of light is a constant, it is just that light travelling through a medium doesn't necessarily spend all of its time travelling in one direction.
I think this is wrong. Regardless of the medium the speed of light is always constant. It seems to slow down because the photons are getting absorbed and re-transmited by atoms. But the speed of light is always the same regardless.
"Light traveling within a medium is no longer a disturbance solely of the electromagnetic field, but rather a disturbance of the field and the positions and velocities of the charged particles (electrons) within the material. The motion of the electrons is determined by the field (due to the Lorentz force) but the field is determined by the positions and velocities of the electrons (due to Gauss' law and Ampere's law). The behavior of a disturbance of this combined electromagnetic-charge density field (i.e. light) is still determined by Maxwell's equations, but the solutions are complicated due to the intimate link between the medium and the field. Understanding the behavior of light in a material is simplified by limiting the types of disturbances studied to sinusoidal functions of time. For these types of disturbances Maxwell's equations transform into algebraic equations and are easily solved. These special disturbances propagate through a material at a speed slower than c called the phase velocity."
As another commented pointed out, you're splitting hairs. The "speed" of light and how fast it is "moving" depends on how you define those terms. The second paragraph of the wikipedia speed of light article has it right "The speed at which light propagates through transparent materials" - which does change.
Here is a relevant piece:
When light enters a material, photons are absorbed by the atoms in that material, increasing the energy of the atom. The atom will then lose energy after some tiny fraction of time, emitting a photon in the process. This photon, which is identical to the first, travels at the speed of light until it is absorbed by another atom and the process repeats. The delay between the time that the atom absorbs the photon and the excited atom releases as photon causes it to appear that light is slowing down.
If the photon is traveling less D over the same amount of T, I am ok with saying the velocity is lower, and it has slowed down.
Nobody would say that they slowed down if they increased their speed as they went through a turn.