To add onto the pedantry, it's only changes in gravity which propagate at the speed of light. A static gravitational field doesn't need to propagate.

OTOH my understanding of the speed of light in a medium is that it’s the result of photons being absorbed and re-emitted, and the speed of any individual photon is always exactly c.

I am an experimentalist and so if a theoretician reads this they will probably tell you that I am wrong...

I think that the description of a photon being "absorbed" and "re-emitted" could be used to describe the picture from the point of view of quantum field theory (which I don't claim understand), because within this theory the photon/electron and even electron/electron interactions are mediated by photons that are created and annihilated during those interactions. Whenever the "photon" exists it will travel with speed c. As light travels through a material it is traveling as a wave of electrons influencing each other, similar to how water waves travel through water, and since these interactions of the electrons pushing each other are formally described by the photons popping into and out of existence I think one could correctly use the language of "absorbed" and "re-emitted".

But personally I think that it can be a bit confusing, because the absorption and emission of light by materials is often used to mean something very different... Absorption more commonly refers to a resonant interaction in which a photon is destroyed and a molecule (or atom, or crystal, etc...) comes into an excited state. The molecule that becomes excited can remain excited for quite a long time (usually picoseconds - microseconds), and the re-emission of the light often comes in a completely different direction and even a different wavelength than the original photon. So using the language of "absorption" and "emission" in this context can also generate confusion,.

Personally when I imagine the propagation of light through a material I think about it in terms of the polarizability of the medium. When the light propagates through a medium, you don't need a "photon". The wave is being carried by the electrons oscillating (these are very small oscillations - unless you are using powerful lasers, then you reach the beautiful world of non-linear optics). The speed of propagation of this wave through the medium depends on how far the wave can travel through the material with every oscillation. There is a nice description of this semi-classical process in the Feyman Lectures: https://www.feynmanlectures.caltech.edu/I_31.html

gravity moves at the speed of light in a vacuum

Hmmm... Always? Maybe some funky things happen as the wave passes by a black hole.