Is there a difference between harmonic motion and _simple_ harmonic
motion?
I always thought "simple" was necessary to distinguish
a simple harmonic oscillator from a _damped_ harmonic
oscillator.
The DHO is quite a bit more complicated. For starters,
it can oscillate at an infinite number of frequencies,
unlike the SHO which has a unique frequency.
But both are harmonic. That's important for several
reasons, including allowing Floquet's theorem to
apply (frequency of response = frequency of drive).
For an anharmonic oscillator, Floquet's theorem does
not apply.