Does it make sense to say, "The speed of light varies?" Some may say right off the bat "Yes, it changes as a wave passes through a different medium." However, I'd like to say no, because when I hear someone say the speed of light, I always think of the constant $c$ (unless the medium is specified to not be a vacuum, but then it isn't $c$ anymore), not the speed of a particular wave. To me, it makes more sense to say something like, "The speed of a particular wave varies." What is the correct way to state this in the professional world? And in general, when professional physicists say "the speed of light," are they referring to the constant or the actual speed of the wave?
I would generally say that most physicists mean "speed of light in a vacuum" when they say "speed of light," and therefore would say that the "speed of light is constant." If it is in a field that often deals with light propagation in materials (optics, condensed matter), people are usually pretty careful to say "speed of light in a vacuum" when they mean it.
Generally whenever some says "the speed of light is a constant," most physicists will assume they mean speed of light in a vacuum.
The speed of light is constant. The reason the observed speed of light seems to slow down as it passes through a medium is because the photons interact with the particles of the medium. The photons never actually slow down, we just perceive their interaction with the medium.