Since I've finished my PhD I've been asked to perform some teaching duties. These duties correspond roughly to the three cases I describe below briefly.
1) So in this one course I was responsible for preparing the contents for classes which were mostly theory on the course subject. I would hand out problem assignments each week,I would discuss them with the students in class but not solve them. In the department I was the only one responsible for teaching this course.
2) On this other course I was responsible for solving problems in class for students. There were other professors teaching the course. Namely, theory.
3) Finally, on this other course I had to supervise students while they conducted some experiments in lab.
For each of these cases which is the designation that best describes my role in class? For example, in 3) I would say that the best designation that describes my role in class is "lab instructor".