Using a textbook slinky as an example, if the disturbance propagates through the slinky from left to right and the particles of the slinky vibrate up and down, does that mean 'wave motion' is also associated with the medium? Since the motion of the wave that we perceive is the combined motion of the disturbance and the medium?
This answer is maybe not the most straightforward satisfactory answer to your stated question, but I think it anticipates ways of thinking that are used in more advanced areas of physics.
There are two pictures of what a wave is.
- A wave is coherent motion in a medium; as time progresses energy moves through the medium and vibrations occur in different locations.
- A wave is a propagating disturbance. It is not made of anything, the word "wave" refers a disturbance which propagates energy from one place to another.
Your question kind of implies that a wave is some combination of 1 and 2. I would say that either 1 or 2 are valid pictures, but you should treat them as distinct pictures of the same physical phenomenon and not reason about both simultaneously.
The advantage of the first picture is that it gives you a clear mechanical model of what is going on at a fundamental level; if you zoom in there are particles in the material, and the particles are oscillating back and forth in tandem -- that coherent motion is a wave. However, the disadvantage is that wave phenomena occur in many circumstances, and there are features of any particular example that will not generalize and can lead you astray if you take them too seriously. For example, light traveling in vacuum cannot be accurately visualized as motion of particles.
The advantage of the second picture is that it is more abstract and general -- wave phenomena occur in all kinds of materials, and so there is no need to specify which specific material you are thinking of, because we can make very general statements about waves that apply to any material. The disadvantage is that it can be hard to wrap your head around a disturbance without a medium, and also sometimes trying to be too general means you miss special aspects of the particular situation you might be interested in (for example, cool behavior like solitons can occur in water but not in light propagating in vacuum).
Here are some more advanced comments not directly relevant for your question but which I could not avoid typing up.
As you progress in physics, you will find there are more abstract kinds of waves that don't have both interpretations 1 and 2, or where it is important to distinguish the two interpretations because they lead to different math.
For example, light is an electromagnetic wave that does not propagate in any medium at all; it is a disturbance without a medium (in vacuum).
When light travels through a medium like glass, its speed changes due to the refractive index. This can be explained due to the interaction of the light with the medium (glass), but the character of the explanation depends on whether you use interpretation 1 or 2 to describe motion within the glass.
For example, one explanation of the refractive index is that the light causes oscillations of electrons around fixed nuclei in the glass, that creates another light wave which interferes with the original light wave, causing a time delay; here the key ingredient is motion of the material (electron oscillations).
Another explanation is that light is made of photons (quantum mechanical "packets" of light), and these photons interact with phonons (quantum mechanical "packets" of sound) in the material. The phonon-photon interactions cause the photon to acquire an effective mass. In relativity, massless particles travel at the speed of light and massive particles travel at less than the speed of light; so this effective mass causes the photons to travel at less than the speed of light once the interactions with phonons are accounted for, explaining the refractive index. These phonons are "particles" describing a propagating disturbance, but the details of the material in which they propagate are not particularly relevant once you know the mass and interactions of the phonons.
For a wave moving through a medium, such as a slinky wave, the motion of the “disturbance” is the motion of the “medium”. Specifically the coordinated local motions of the medium that propagate through the material at the wave speed constitute the disturbance/wave. For a transverse wave on a slinky, it is the transverse local oscillations of the medium which collectively produce a wave that travels along the length of the slinky.