Diaphragm singing

Now, take one of these deep low breaths and and hiss it out on an 's' or 'sh' sound. You should not feel any tension in your throat as you do this. As you are hissing, concentrate on the feeling of tension below your ribcage and try to control the air by pushing back against the tension you feel in those muscles. Congratulations; the tension you feel is your diphragm contracting and expanding. This sensation is what you're looking for to support the sound you make while you are singing; try taking in some low breaths and send out the air on a note, keeping in mind this sensation and trying to keep your throat as loose and relaxed as you can. Then try singing some scales and intervals.

This probably will point you in a better direction than I could. Are you asking why they teach you to sing vowels differently from the way you speak them?

Put your hand on your throat and swallow. Feel how the larynx moves up, and then drops back down? You want to try to keep your larynx in that lower position (this is often referred to as "opening your throat"). If you're doing it correctly, you should feel this sensation that your throat is HUGE and open (keeping in mind that you don't want to strain your neck muscles trying to force your larynx down).

90% of healthy singing (classical or otherwise) can be boiled down to the twin concepts of relaxing everything above your shoulders and working from below your ribcage.

OMG coffee burns when it's in your nose

I have a question--what is the high, throaty voice I use to get this breathy, Martin Gore-y sort of tone?

I can't figure it, but there it is.

Tie a belt or rope around your waist loosely. Breathe from down there until you expand the rope/belt AND you feel your sides/back expand. Hold that out while you sing the phrase. Then collapse.

To strengthen the diaphragm, lie on your back flat, and but books on it, and breathe.

Diaphragm breathing is how you breathe when you first wake up in the morning, when you are sleeping. It is from down deep. That's what you are going for, so pay attention when you wake up in the morning to how you are breathing.

Hope that helps.
$50 please.

I have always been so undisciplined about this.

My first voice teacher told me that the best thing for me would be to do abdominal exercises.

But how does any of this explain why there are so many, uh, hefty singers who achieve greatness without ever having visited a gym?

Their core is still pretty strong is what I have been told.