Thanks
@russ_watters and
@lychette.
So, sounds like you guys are saying the defining property of a medium, in this sense, is that the medium is involved in the movement of a wave itself (a propagated displacement), such as water or air waves move through a medium of water or air because the water or air molecules are actually moving (up and down or back and forth) instead of just something it moves through.
In which case it seems to me that space-time would be the medium for gravity waves but not light waves, but no one has said that (at least in a way that I understand).
Seems to be required for propagation, affects the propagation of light (as ether was claimed to do).