This doesn't have anyting to do about the issue of the speed of light in a media vs 'c'.
This issue has to do with how velocities are measured. People who rely on coordinates are usually the ones confused by this issue.
The physics is actually very simple - when you use local rulers and local clocks, the speed of light over a short path is always 'c'. This is true even in accelerated coordinate systems, though one has to pay attention to the "short" part of the above statement.
People often use coordinate systems to measure the "speed of light" and think of the speed as being measured by the rate of change of a distance coordinate with respect to a time coordinate. This is a fundamentally different definition, for it uses coordinates to measure the speed, rather than local clocks and rulers.
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An analogy might help. Suppose someone tells you "We have a naval ship, and it always travels in the sea at a constant velocity". Now,if you set up a lattitude/longitude coordinate system on the Earth, and measure how many degrees of longitude the ship moves in one hour. you would find that the ship moves faster and fasater as it gets closer and closer to the north pole. This is the coordinate based definition of speed - one measured the coordinate of the ship (its longitude) and how fast it changes with time.
However, someone actually on the ship, measuring its velocity relative to the water, fnds that the speed of the ship really is constant, as you were informed of in the first place. So the confusion arised because of differences in defintions.
For this example, we are ignoring the issue of ocean currents - assuming that they are negligible. In relativity, of course, there are no "ether currents" to worry about.
In our naval example, the issue is how one defines distances and velocities. Using coordinates, as one gets closer and closer to the north pole, a minute of longitude (a coordinate measure) is not the same as a physical nautical mile in one's local coordinate system. The curvature of the Earth is responsible for this confusion, just as the curvature of space-time is what makes the "coordinate speed" of light vary.
One can set up a metric that gives physical distances on the Earth as a function of changes in coordinates - i.e. distances as a function of \delta lattitude and \delta longitude. This metric for the curved surface of the Earth is conceputally the same as Einstein's metric for space-time, except that Einstein's metric gives the invariant Lorentz interval, rather than the distance.
One can see that when the metric coefficients are diagonal and all equal to 1, physical distances correspond to coordinate distances.
You might ask "Can we set up a coordinate system so that the metric of the Earth is everywhere an identity matrix"?. The answer is no, and the reason this can't be done is because the Earth's surface is curved. Relativity says the same thing about space-time - in the presence of curvature near large masses, you can't make the metric coefficients equal to an identity matrix everywhere, though at any particular point you can "normalize" them so that coordinate distances are equal to physical distances (ruler distances) and coordinate times are equal to clock times.