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.

Honestly I kinda laughed when I saw the response because it reminded me of the classic nerd flexing (that happens especially when conveying something to laymen) where two nerds go at it saying the same thing but using heavier and heavier jargon as a means to flex (increasingly losing the laymen) rather than clarify or admit that language requires inference and the whole show is ironically just a demonstration at accurate communication or else one would not have been able to "um acktually" the other. (I think a lot of us have been guilty of this, often unintentionally, but we can still laugh at ourselves)