In general usage, "sound" refers to our perception of the vibrations of particles (atoms, molecules) in some medium, typically air or water, though sound waves can travel through any medium.
Vibrations are produced whenever objects cause the particles in the medium to oscillate, e.g. clapping your hands together, beating a drum, or yelling.
Sound can be measured the way it is heard - just as air molecules begin to vibrate in the presence of a vibrating object - the air molecules can cause other things to vibrate (such as our eardrums, or a part of a microphone) and we can quantify the level of sound by measuring the motion of the detector in any number of ways.
At a deeper level, sound waves are the "Goldstone modes" corresponding to broken translational symmetry of the surrounding medium.
Let me try to bring this a little more down to earth. Think about a parcel of air, large enough to contain a lot of air molecules, but not so large that there are sizable air currents swirling within. We can think of this in an ideal situation as being a fluid of uniform density. This fluid has translational symmetry - if I translate it a little in any direction, it looks more or less the same "in the bulk". Translational symmetry is a continuous symmetry, so I can squeeze the air in a spatially-varying way. It turns out that if I do this in a wavelike pattern with a long wavelength, this is an excitation that costs very little energy, and hence will be important to the physics. These waves are precisely sound waves. This probably doesn't make much sense as I've written it, so I may try to edit this a bit later on if anyone is actually interested in this point of view.
The first thing to understand is that dB is a logarithm of a ratio. More specifically, a logarithm of a ratio of power/power (which is why dB has no units). A Bel is the base-10 logarithm of a power ratio. A decibel slices each Bel into ten parts.
Power is power. It does not matter whether it is sound or electricity or whatever. And dB always refers to a ratio of power.
In most sound references, "dB" means "dB in reference to (as a ratio over) 'X'." Where 'X' is a reference level. I think the reference used is the sound pressure which is the threshold of human hearing. So a sound with 10 times the POWER of that reference would be 10 dB. A sound with POWER 100 times the reference would be 20 dB.
The answer to your question is: It can be. Sound intensity (if expressed as a pressure) and Sound Pressure Level (SPL) are the same. But neither is power. Sound power is proportional to sound pressure squared. Thus, if you increase sound PRESSURE by a factor of 10, you have increased POWER by a factor of 10^2 = 100. These two actions are synonomous, and result in a 20dB increase. Note that I did not explicitly state what increased by 20dB in that last sentence... because it is ALWAYS power.
So, you need to watch your units, and make sure the other guy is watching his. If he doubles his POWER level, that is a 3 dB increase. If he doubles his sound PRESSURE, that is a 6dB increase. If he doubles his sound INTENSITY, you need to nail down whether that is in units of pressure or power.
I hope that helped.
Best Answer
The wavelength of sound waves in a solid is much greater than the dimensions of atoms. For example the speed of sound in steel is 6100m/sec, so the note middle C (262Hz) has a wavelength of about 23 metres. Sound waves are collective motions of a vast number of atoms, and it isn't especially helpful to think of them as being generated by atom scale phenomena.
Because, as you mention in your question, atoms don't have a sharp edge, in a solid like steel atoms can be pushed together slightly and pulled apart slightly. This gives the solid some elasticity, and this elasticity allows compressions waves (i.e. sound waves) to travel through the solid.