In my view, a lot of this discussion gets sidetracked by ambiguous language.
If two people sing an octave apart -- or a 5th apart -- it sounds lovely, doesn't it? Would you call that "distorted", from an aesthetic standpoint? Probably not. But compared to a single voice, it is "distorted" from a technical standpoint.
It's weird, isn't it for something that sounds mellifluous and pleasing to also be distorted? That weirdness comes from the ambiguity of the aesthetic meaning of "distorted" and the engineering/measurement meaning of "distorted." But they're not the same thing.
Is a tube amp distorted, then? Technically, "yes," and aesthetically, "no" -- if you like that kind of aesthetic effect.
Here's something useful from Tung Sol:
"The harmonic content of an overdriven tube amplifier consists primarily of 2nd order and 3rd order harmonics with some 4th order harmonics. The harmonic content of an overdriven transistor amplifier is primarily 3rd order with suppressed 2nd order harmonics. 2nd and 3rd order harmonics are the most important from a viewpoint of electronic distortion. Musically the 2nd harmonic is an octave above the fundamental and is almost inaudible, yet it adds body to the sound, making it fuller. The 3rd harmonic is a musical 12th. Instead of making the tone fuller, a strong 3rd harmonic makes the tone softer. The odd harmonics (3rd, 5th, etc.) produce a "stopped" or "covered" sound. The even harmonics (2nd, 4th, etc.) produce a "choral" or "singing" sound. Adding a 5th to a strong 3rd harmonic give the sound a metallic quality that gets annoying in character as the amplitude increases. A strong 2nd with a strong 3rd harmonic tends to open the "covered" effect. Adding the 4th and 5th harmonics to this gives an "open horn" character. The higher harmonics, above the 7th, give the tone "edge" or "bite."
https://www.tungsol.com/html/faqs14.html