Indeed, to support your point, I had a teaching assistantship for my first two years of grad school, meaning that I worked on an as-needed basis for things like supervising labs, proctoring exams, grading papers, etc. It was funded by the department. Where I went to school, the TA workload was pretty light.

Then I got a research assistantship, which was funded by my advisor's grant. I meant that I got paid to do my thesis research, and no longer had to teach. This was a precondition for joining his group.