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You are way over complicating this in avoiding just googling the actual definition of sound in a physics context. It's acoustic waves in a medium. Acoustic waves are adiobatic compression/rarefaction waves.

All your examples are just sound. There's no difference if the medium is all gas, all tennis balls, or a mixture of both along with some very confused corgis.

The medium, and whatever objects it exists as, are not sound itself. The notional particles of sound waves are called phonons.

Propagation of transverse waves in the electromagnetic field is what we call light, radio, and other electromagnetic radiation. There's also constraints of symmetry for how the electric and magnetic portions of the field relate to each other. The notional particles of these waves are photons.

To address your last point, it would help to stop thinking of waves as platonic objects with their own independent existence as objects, and instead see them as patterns of activity/interaction within ongoing dynamic systems.

All of your examples are simply sound. There's no confusion in this. And yes the definition of sound still requires a medium.

> What if we replace atoms with electromagnetic balls exhibiting similar bouncing properties, would it be good enough to count waves in this medium "sound"?

What do you mean by "electromagnetic balls"?

> My point is that if we can substitute the medium carrying the waves, than we may as well remove the medium from the definition of sound.

Again, sound, simply by definition, is a compressive wave. Compressive waves can only happen in media that can be compressed, which rules out fundamental fields like the EM field or space-time. Atoms may not be entirely fundamental to sound, but matter is - you may be able to have sound waves in a neutron star for example.

I wonder if it's possible to talk about sound waves inside the radius of a black hole - that I'm not sure about.

I would call it sound in tennis balls. They're still using the same fundamental mechanism of pressure to transmit sound. But light isn't sound because the mechanism of propagation is fundamentally different, not just at a different scale or with a different medium. It's subject to different laws and behaves qualitatively differently. For example, it can be polarized.

I wonder about a neutron star though? Is that still subject to this theoretical speed of sound limit, or is it only atomic substances?

We can go even deeper and ask how we'd call sound-like gravity waves in spacetime? Some sort of oscillating system of heavy stars can produce a periodic gravity wave that we'd call sound probably.

No, we don't call that sound. Those are called Gravitational Wave. Admittedly this is a bit confusing vs Gravity Wave, but we appear to be stuck with that one in English now.

That's sound. Very lossy sound, presumably at a very low frequency, but sound nonetheless. There's a physical medium, and a signal is being passed through it via tennis ball pressure. If you had a sufficiently dense field of tennis balls, you could visually observe the wave moving across it as the field compressed (in the direction the signal is traveling) and then uncompressed, cyclically.

A good trick to telling the difference is thinking about the direction of the frequency. Frequency is a back and forth movement. If it's going back and forth in the direction the overall signal is moving (like a tennis ball going faster towards the destination and then more slowly or backwards, or like the wall at the source of the signal vibrating towards the other wall), that's a physical wave. If the back and forth movement is happening in an entirely different plane, that's like an electromagnetic wave.

So just to be clear, as it seems the other poster is quite confused, whether waves are "physical" or not is independent of whether their traverse waves or density waves. Sound is an example of density waves. The waves at the interface between the ocean and the air are gravity waves. Both are physical.

EM waves are physical as well, they just are in the EM field itself, rather than having a medium like matter.