harmonic distortion and frequency response

I'm not sure I understand your question or what the designer was trying to say,but let me try confusing the issue from a different direction :)

Go to a piano and hold down a key(without making it sound),and then play the note an octave below it;the higher note will ring as an "overtone".

Make a guitar string vibrate,and you sound a note,the overtones that will sound,in combination with the original note can be heard by cutting the strings vibrating length in half,the first overtone,and cutting it in half again,the second overtone. Further,"difference tones" underneath the tone will sound if you play a note that vibrates at 80 vibrations a second,it will generate vibrations and 40,20,10 etc. By the same logic,the overtones double(strike a 110;the overtones are 220,440,and so forth.

These combinations of overtones,and how much amplitude they have determine the timbre or tone quality of something,and more,say a violin and a cello will(any two instruments) will vibrate each other's overtones and difference tones,and combine this with 80 instruments,and you see how orchestration works-kinda.

The 1st,3rd,5th,and 7th overtones are "odd".
The 2nd,4th,6th,and 8th overtones are "even"

(More? Seashore,Carl. The Psychology of Music.)

Some tube amplifiers(including amps) have a more complete overtone series,so even their distortions don't sound bad.

Some solid state amps(including pres) are more likely to overempasize odd overtones. *Also,the 7th overtone(correct me if it's the 5th or the 9th is inherantly flat of pitch,so if you have too many 7ths and too few 4ths and 6ths,it MIGHT sound out of tune,but my ear is not that good. Beyond that,I yield to people who know why.

If a cable can effect the transmissions of signals enough to effect these proprtions of overtones is something I do not know. If could be sales puffery,but I don't know that to be a fact.

I think you are asking about a few things.
3rd order linear & non-linear distortion: this is very often encountered when a SS amp is pushed beyond its energy providing capabilities (the amp is "clipping"): the resulting sound will be compressed (you tell by the amplitude) and one sonic byproduct is a generation of 3rd order harmonics of the original signal -- which are not part of the signal fed into the amp. These in turn are fed into yr speaker which will reproduce them.
Humans find these sounds strident, unnatural. We are used to accepting 2nd order harmonics: hit a resonant item and you'll hear the basic sound (fundamental) and other sounds, higher in pitch -- these are 2nd order harmonics. Musical intruments naturally produce 2nd order harmonics; some of these are captured in recordings and are reproduced, allowing us to distinguish between different instruments, voices, etc.

Such distortion, at high amplitude, is rare in normal operating conditions.

(BTW, the sine waves used in testing, are "pure" waves in that they don't contain harmonics -- i.e. they are unnatural sounds, but useful for test purposes only.)

Another question I think you;re asking concerns the sound emanating from the speakers: frequency response and "tonal balance". I don;t know, but assume, you use these expressions interchangeably; however, the ultimate sonic result in a room is of course the product of a system as a whole. In this case, there are many factors that determine the "quality" of the sound; one of these is frequency response -- but, within limits (~+/- 5db spl) it;s not the most important one; phase, power response, extension... are just a few.
Cheers