Speed of light and distance Our measure of distance (the meter) is defined in terms of how far light in a vacuum travels in a specific time.
When light travels through another medium, we say it travels at a different speed. Why not say it travels at the same speed, and the distance in the medium is different that the distance outside? Is it only because the 2nd explanation feels less intuitive than the first, or is there more to it? 
 A: While our definition of the meter is based on the speed of light, our definition of length is not. Naming and defining a unit of measurement is entirely different than defining a physical quantity. The most natural way to define distance (known by physicists as proper distance) is the measured spatial separation $\Delta r$ between two points in space. Rigid objects are measured in a frame of reference where their speed is 0. We cannot say the speed of light is constant in different media, because this is simply untrue. Phycisists have the choice to choose units, as they often do, but they do not have the ability to completely and arbitrarily redefine physical quantities. Physical quantities such as mass, time, length, etc arise naturally in studying nature, and are not mathematical consequences of classical theories. Moreover, while certain modern theories explain the existence of these quantities, they do not define them.
A: When travelling in a dielectric light isn't light. It intracts with the medium to form a composite system that has an effective mass and therefore travels slower than $c$. If the interaction is strong, as in a BEC, the interacting system can be described as a quasiparticle called a polariton. This isn't useful for weakly interacting systems like most transparent solids, but nevertheless the same principle applies.
The point of all this is that your going in point is wrong. Light doesn't travel at a different speed in a medium, so we don't have to worry about redefining distance.
