[pa7x1]

The basic difference is pointing that there is something more fundamental and it doesn't have to do with physics. It has to do with math, with calculus, and with approximating complex functions via perturbative expansions. This is very general and applies to other kinds of systems, not necessarily physical. That's the key insight.

The first relevant term is quadratic in the potential around a stable equilibrium but linear in the force. That's why they are called linear harmonic oscillators, I thought that was obvious in my description.