All sound is motion, but not all motion is sound.
The first topic to discuss would be oscillation. In theory, one can use the Fourier transform to decompose any motion into the sum of oscillating motion. So one could argue all motion is oscillating. However, the devil is in the details. There's a lot of cases where breaking that motion up into oscillations only muddies the waters, making the laws behind the motion harder to see.
Looking deeper into how sound behaves on such oscillations, sound implies wave motion. There are particular equations describing a wave. If you can describe motion as a wave, then it can be thought of as sound.
Sometimes this is easy. Sound traveling through the air as a pressure wave is extremely well modeled with a wave equation. Everything you know about sound makes a lot of sense when describing oscillations in air.
Sometimes this can get hard. Gongs, for example, are a very non-linear instrument. Some of their behaviors are well described as nice wave equations. But they also have some very interesting non-linear behaviors which give gongs their unique shimmer. We still think of these in sound terms, but the equations start to get more complicated.
In many cases the concept of "sound" just doesn't apply well at all. One very direct example of this is in the shock wave in front of hypersonic vehicles. Pressure, as it pertains to pressure waves that make up sound, pushes out equally in all directions. At those extreme speeds, pressure starts to become a directional concept -- the pressure you see in the forward direction is different from the pressure you see in the backwards direction. This changes the behaviors of the air by quite a lot. There are many properties of this shockwave which are so hard to describe in sound terms that it's really best to think of them as "not a sound." I find this example useful because we start with something very "sound" like, vibrations in air, and find that when we pushed it to the extreme, it stopped being very sound-like.
Another example might be the motion of a ball thrown in the air and caught in a mitt. I'm actually unsure whether I could phrase that problem in sound terms or not. But if I did succeed, it would be so contorted that I'd probably need a full novel worth of words to describe everything that happened. However, if I don't think of it as a sound, I can capture most of the motion with nothing more than "F=ma". Even if I could think of that motion as a sound, it would only hurt me to do so. It is more effective to say that motion isn't a sound.