Some CS departments are known to be "theory" heavy whereas others are more "applied" heavy. In theory heavy departments many students do not get exposed to the day-to-day details of writing perfomant code and they have the feel of being an offshoot of the math department as opposed to engineering.

That's one of those decades-long debates that continues. Is CS actually "science"? Are developers actually "engineers"? Is a person in charge of a Kanban board an "architect"?

There are absolutely computer science jobs, but in general, a casual distinction would be what side of the research/implement line you fall (though in truth, it's probably more of a 2D graph than a hard line).

Computer Science is the study of computing

Building software is just one application of computer science

It's a bit like the difference between someone who is a psychologist helping people in therapy, and a psychologist who studies psychology in a lab. They require similar educations but they are very different professions