What is the difference between the breathing that causes your shoulders to rise and the breathing that enlarges your stomach? Why is the first type considered wrong?
When your shoulders rise, you engage your ribcage with a mixture of breath work and other mechanical business. Both disrupt the connection to the diaphragm where you want to anchor the base of your breath control and resonance.
Controlling breath with your rib cage is like trying to run with your calves. Of course they are involved, but you don't want to disconnect from the thighs. Or even the arms: if you are going for a sprint, immobilizing yourself above the hip is not going to your full potential. Now singing is not a natural endeavor: not doing anything with your arms when running still beats flailing them about uncontrolledly. But in the long run you will want to arrive at a stage where your breath control acts throughout your body in harmony and elasticity. Part of that overall goal amounts to a coordination in relaxation and breathing muscle coordination that more or less runs under the name of "breath support".
The kind of tension leading to drawn-up shoulders is running counter to that goal in several respects.
We are talking about two kinds of breathing: Costal breathing, the lifting of the ribcage, and diaphragmatic breathing, the manipulation of the diaphragm, which is the most powerful muscle in the body and which sits just below the lungs.
Proper singing involves a combination of costal and diaphragmatic breathing.
However, raising your shoulders and raising your ribcage, by itself, is counterproductive. Raising the shoulders adds a great deal of tension to your throat and larynx, which produces a strained vocal sound. You should never permit your shoulders to rise when you take in a breath to sing.
Learning diaphragmatic breathing is the most important technique. It has to do with learning to manage the way that you fill your lungs with air so that you have enough air to sing long phrases, to produce a smooth, consistent vocal tone, and to control how much vibrato you use. We call this "breath support". Breathing from the diaphraghm, and supporting the air pressure in the lungs from the diaphragm, is the most efficient way to sing with control.
Once you learn to sing and support the breath from the diaphragm, then you also learn to add the use of the costal muscles to supplement diaphragmatic breathing.