Personally, I would never thing that "teach at a University" implies a professor, but that is probably due to the number of non-professor instructors I have had for classes. Also, graduate programs typically pay the students for either teaching or doing research in a lab, at varying ratios depending on the program. Where it is required, if you don't teach, you don't get paid, thus if the question is "What do you do for a living?", teaching or working in a research lab would be the honest answer, because that is what you are paid for.