Are there different ways of breathing for the singer? 

Yes, there are three. The shortest breath, generally used for speaking, reaches but superficially to the upper part of the lungs near the collar-bone. The is called clavicular breathing. 

The second is lateral breathing. Here we dive considerably deeper, reaching the parts of the lungs which lie between the armpits and the waist. Though a good deal deeper, this is still not sufficiently deep for singing. 

The true singer's breath is called diaphragmatic breathing. Here the lowest extremity of the lungs is reached, this kind of breath being the only one to fill the lungs completely. 

How are we to know that our breath is deep enough to fill the lungs? 

By comparing it to a yawn. The singer's breath must be as deep as the breath we take in yawning. 

—Blanche Marchesi, The Singer's Catechism and Creed (1932).