They have them in universities too, apparently for parents with babies, providing quiet spaces for lectures. However, during medical school, plenty of students used them as rooms to literally cry in during exam periods (myself included).

Not really a ‘hmmm’. Not sure how widely-used the term is in non-church settings, but this is the room where you take children who start crying during a church service. It’s intended to be a place for them to calm down and/or the parent to get control. It can also be the room where nursing mothers feed their babies.