Robert, you're right, every singer produced overtones. I define overtone singing as a singing technique that intends to make the overtones audible as separate tones with their own musical purpose. But there is indeed more about that in classical singing. My new complete "singing phonetics" will be published shortly. It's related to what you say. Recently I presented my "choral phonetics", which is part of the more comprehensive singing phonetics, at the International Stuttgart Voice Days. Find a short version on my website (you'll find an english pdf at the bottom of the page): http://www.oberton.org/chorphonetik/
At the 5th International Conference on the Physiology and Acoustics of Singing in Stockholm 2010 I presented the two main differences between classical formant tuning and overtone singing:
1. in formant tuning the singers are usually not aware of individual harmonics,
2. in overtone singing you combine two formants to build a double-resonator in order to separate a single overtone acoustically.
Find more here (use Google translate for now, I am working on translation of my whole website): http://www.oberton.org/obertongesang/was-ist-obertongesang/
If you're interested in the benefits of overtone singing in classical singng I'd be happy to introduce you to the fascinating world of overtones that I am exploring scientifically and artistically since 30 years. (BTW, Anna-Maria presents in her video some of my advanced singer's exercises, and the software used for visualizing is Overtone Analyzer, which I developed together with Bodo Maass).