Who thinks you can sing from the diaphragm?

Well, the "Singing from the Diaphragm" is an old world concept that was probably taught so that the person will not squeeze their throat. The correct terminology is "Good breath control". It means that you have to breathe properly. Obviously, the sound does come out of the throat, however, it shouldn't feel like it comes out of the throat. Good breath control involves taking "low breaths" which is necessary for proper support. The "support" term is used in regards to "resistance" or "holding back the air pressure". The air pressure must be held back with the lower body, NOT THE GLOTTIS. Holding back with the glottis is similar to feeling the sound coming from the throat. Now, there have been cases reported in which people use the "diaphragm" to produce a fake vibrato. I do think that they're using a different muscle.

Vocal cords are located in the throat. Air goes in and out of lungs, existed by diaphragmatic muscles. A singer can be taught how to take deeper breaths to support sound production, but it is the control of the air flow that allows this--not the actual breath itself. Let the air out at once, and you are out of breath as if you had not bothered to take a preparative breath to begin with. The term "breath management" is probably a more accurate term than "breath control" to describe the process. What is very important to remember is that singing is not all breathing. Proper use of the resonating cavities are vital. Posture and position of head, neck, jaw, and tongue are important. There is a LOT of physical coordination involved in healthy singing.

Yeah Liz, I've often thought it was an odd expression, but it's one of those terms that's been around so long it will never go away. It's a misnomer of course, but I think most people understand it intuitively.