Elementary Excitations and Phonons

I am currently reading "An Introduction to the Theory of Superfluidity" by Khalatnikov and trying to reformulate my understanding of phonons, which used to be something along the lines of "phonons are quasi-particles necessary to describe acoustic waves in the same way photons describe light waves." On p.4, before discussing the energy spectrum of liquid helium II, the author introduces elementary excitations and quasi-particles. So far, I find the reading quite sophisticated and lacking a clear logical structure. Thus, I have summarized my understanding in the following manner, but I feel that it might be inaccurate - more specifically, I am not sure if we can draw a parallel between some classical concepts and those in quantum mechanics. Essentially, my question is: where is the mistake in my analogy, if any?

$1)$ From Classical Mechanics, at $T=0$ all atoms are at rest with $KE = 0 $ and minimum $PE$.

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$2)$ Atoms can only undergo small oscillations around their equilibrium positions.
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A short digression: In Classical Mechanics, we introduced the concept of normal modes for a system of coupled pendulums. When you have two coupled pendulums, they can oscillate in phase and out of phase - those are the only two unique ways the system can oscillate, and they are called normal modes. Mathematically, the description of these two motions comprises an eigenbasis - namely every other motion may be described as a combination of these two. End of the digression.

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$3)$ Back to our lattice with vibrating atoms, we also want to break down its vibration into some normal modes - not vibrations of single atoms but rather collective vibrations of many atoms. We name the normal modes of a vibrating lattice / liquid helium elementary excitations. But in doing so, we view these elementary excitations as a quasi-particle moving through the body with a definite energy and momentum. Those elementary excitations behave like different quasi-particles, hence we introduce phonons, maxons, rotons, and etc. to describe the atomic oscillations - though rotons and maxons are used in the context of a liquid helium.