Yes, that seems to be the problem here. I thought I made it pretty clear what kind of "beats" I was referring to, but I also acknowledge that the term "beat" can include mixing. A clear example was already mentioned, to wit, the BFO, which of course is a mixing operation.
I do disagree totally with whoever thinks mixing is done in the ear to any audible extent. The lowest audible sound would have to be at the sum frequency, i.e. at twice the t-f frequency, which it clearly isn't; or it would have to be a very high harmonic of the difference frequency, which still would be at a very low frequency, near the lower end of audibility, which again is not at all what the tuner hears. So please, folks, forget about nonlinear ear response!