You're right, partially. Cow stomach is included in traditional cheese-making, but it is labeled as dairy because the majority of cheese is made of milk, which is dairy. Just like how a meatloaf is called "meat" loaf even though there's a lot of egg, onion, spices and breadcrumbs. The product is still mostly meat.
Vegetarians don't eat anything that caused an animal to die, so the types of cheeses you're talking about are not considered vegetarian-friendly. There are other types of enzymes not derived from animals that curdle milk into cheese just as well. These types of cheeses are considered vegetarian-friendly since no animal had to die for the enzyme.
Of course, for vegans like me, both milk and egg products are gross, unnatural, harmful to animals, harmful fo the body and harmful to the environment, so we avoid them!