[clay_to_n]

I don't think we're hallucinating the overtones here - with an EQ (like in the link I believe) you can clearly separate the high and low freqs, and they sound like different words. So it's overtones baked into the audio file.

[Nomentatus]

But what you're describing is exactly what you'd expect if you were dealing with overtones, actually - one tone of the two actual tones that create the illusion of the overtone has to be substantially lower than the other, always. If you knock out one tone by dumping some frequencies, the overtone vanishes, and those who were hearing it now hear something else. Those who never heard it, hear what they heard before.

The basic idea here is that if the harmonic match of the two fundamental (actual) tones isn't exact, then some listeners will hear the overtone, but less sensitive (or perhaps more refined) listeners' brains won't hear it. And many conditions will make it easier or harder for the listener's brain to (falsely but vividly) infer the overtone, including a lot of acoustic reflection (my bathroom example.) Completely knocking out one of the two fundamental tones will change what's heard by some - since now everyone will be hearing the same thing.

Similarly, shifting all frequencies up or down, even if all the information is preserved, can cause everyone to lose the overtone since the overtone is now at a frequency above (or below) what we can hear, so the brain doesn't hallucinate what is beyond it's capacity.