Hi Adrien, what an insightful and comprehensive question.
To be more specific? If you whisper then slowly engage the voice and then work towards a shout, how do the abdominals work and do you ever really have to think about it?
To be honest, the back and side muscles, I have never been able to isolate when playing but I am sure they must be engaged and I guess that goes for internal and external as well. I think we can over complicate things on this level.
We have done nothing up till the end of the Foundation Level that requires any kind of serious abdominal contraction, other than playing loudly for a long period which involves emptying the lungs. That being said, the idea of how much “work” of the abdominals is actually required is massively over estimated.
Playing a loud, long double high G does not require MASSIVE abdominal exertion unless the oscillator is pinched thus requiring force to make the vibration happen.
Yes, the “Belly have to get in” does happen when supporting airflow especially at a loud volume, just the same as when talking loudly. Overdoing this can create extreme negative tension.
I will explore the appogio vocal technique, I have never heard of it.
All the best,