This noise will show up in the distortion spectrum, as a raising in the noise floor, as the noise is not harmonically related to the sound being produced or the harmonic distortion created…
I'd have defined noise as
“an output product uncorrelated to the input signal” and distortion as
“an output product that bears correlation to the input signal”.
On this basis, I'd be inclined to suggest that any additional unwanted output product must correlate to the input signal (even if its quality is noise-like i.e. fairly wide band). While not thinking it through too deeply, I might suggest that this phenomenon could be what some speaker designers term “cone coloration”.