The idea behind speaking from the diaphragm is to ease your vocal folds into speech, almost like you're tricking them so they don't tense up and block and make the stutter harder. It also helps you have more control over your breath, which some studies have found to be different among people who stutter... Over time you're supposed to go from (very breathy and not really natural) to (a bit breathy and more natural) to (not so breathy, still pretty natural). Singing uses different networks in your brain and involves more control of your breath, so some methods rely on that to make speech more fluid. Something else that focuses on breath is mindfulness (it teaches you to be aware of your breath and of your body, and some methods have been developed to help people dealing with chronic pain, for example). Do whatever works for you and don't beat yourself up if it doesn't.