In the matter of gravity vs. gravitational waves, I've always found it easier to think of (static) electric fields vs. EM waves.
Think of a static field "going out" from a charge. It isn't really going out. The field lines don't have "ends" that travel out at the speed of light. That's because they don't have ends at all. They all end at charges, and they stretch as far as needed. Charge is never created, it's always + - dipoles that are created (like electron-positron pair creation) so there's no problems with field lines that go out from an electron and go to the limits of the universe and end on some + charges out there. They've been stretched for that long, since the Big Bang. We don't think of how fast they move. They've been there since the beginning when the universe was small, and now they span it completely. A billion light years is nothing.
Similarly with static gravity. Mass-energy cannot just appear or disappear, so the field lines never have ends that have to move outward. They're always connected to mass and energy far away, and have had since the Big Bang to do so. There's no point in asking how fast static gravity moves. It's just "there" from here to the edge of the universe.
If you start to suddenly move, with respect to an already established static gravity or electric charge field-line, the field DIRECTION moves immediately with you, and so does the direction to the source. That's just Lorentzian relativity. The speed of light is not being violated. A source a billion light years away would suddenly start to look like it is moving, but that's because its field is already out to where you are, and the field where-you-are, tells you. It changes direction when you move. It responds immediately to relative uniform motion, via the mechanism of the field that is already extended to each.
But if the static or gravitational CHARGE moves (accelerates) then there is a "kink" or update that moves out from it at c. You don't see this at all from far away, until time d/c has passed. That's the EM wave or gravity wave. It's not a relative thing between source and viewer, because acceleration is not "relative" in relativity. You can't pretend that the observer accelerates and the source does not.
The Lienard-Weichert potentials for EM have two terms for this reason. One is the static one that depends only on relative velocity and points at the source (so long as relative velocity has been constant for long enough). The other one shows aberration (does not point at source), retardation, and is a disturbance in the field due to source acceleration (not observer acceleration).
The slow dance of Sun and Moon are a mix of both effects, in the near-field. The static effects point right at the sources, and are due to fields that already extend to infinity. They have no "speed." However, the second order effects due to small amounts of source acceleration (from orbital acceleration) are tiny, but they are genuine gravitational waves, and they move outward at speed c. They are retarded and would show aberration.
For a vibration to exist, something has to vibrate. In the water vibrations it is water molecules. In light it is electric and magnetic fields that vibrate.
In gravitational waves it is the space itself that changes , (x,y,z,t) between points becomes larger and smaller and the wave is seen as a distortion of the ring of partilces, the image is of one wavelength passing.
The mainstream gravitational theory is General Relativity which assumes that what we called gravitational field is an effect on the space time fabric created by the presence energy and masses. The space itself is constructed by the existence of these. Gravitational waves arose from the solutions of the differential equations of general relativity for specific boundary conditions. The "medium" you are looking for is space time itself.
Best Answer
Gravity waves are a term of art used by people who study wave behavior in liquids or gases in which gravity and a free surface or a density difference are present. Gravity waves in the surface of water are the large, fast-moving ones stirred up by strong winds, that surfers ride on and that can crash over and capsize large ships at sea. In this special context, they have nothing to do with waves in the fabric of spacetime.
(The other type of surface waves in water are capillary waves, in which the restoring force that strives to smooth them out is not gravity but surface tension. These are tiny waves that move slowly, and are produced by things like wiggling insects floating on the surface of the water.)