Given that if you perform a local experiment the speed of light is always "c" , regardless of gravity, whether you're deep in a gravity well or far outside it, what do people mean when they say "the speed of light varies"?
What they mean is that in some particular kind of coordinate system, the rate of change of the position coordinate with respect to the time coordinate isn't always the same.
For one, I don't mean that when I say that.

Instead, I mean that for *any* "universal" coordinate system (that uses single, non-local standards for length and time) the speed of light isn't *everywhere* the same. Only for a particular coordinate system that is kept very small, or one that has "elastic units", the speed of light is "always c".
This really has more to do with the behavior of coordinates than any property of light itself.
It's a bit like saying that naval vessels "move faster" when they are near the north pole, because the rate of change of the [longitude] with respect to time is greater.
The Earth is a physical object to which those coordinates relate. Thus, do you mean that:
1. there is a physical equivalent to the Earth, but which we cannot see?
or
2. that it's an artifact of using a certain kind of coordinate system?
- if 1.; what is it?
- if 2.; then Einstein's light bending calculation was based on a mere artifact. Then how do you explain the bending of light?
Harald