For each of these cases which is the designation that best describes my role in class 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".
Teaching Fellow (which is a title usually reserved for an instructor who is also otherwise affiliated with the institution as a student, usually graduate), alternatively, I have seen a few graduate students who taught classes in their own right and were given the title of Lecturer, which is analogue to "adjunct" at some institutions--generally, though, even at institutions that use the title "adjunct" and not "lecturer" for their adjunct professors, graduate students who teach courses in their own right will usually still be referred to as a "lecturer". The reverse, as far as I know, is not true though--at institutions that use "lecturer" over "adjunct" or where both "adjunct" and "lecturer" are in use, students that teach are not called adjuncts, but are usually identified as "instructor" or, though rarely, "moderators", or even, the difficult-to-cv pseudo title of "Course led by...".
Teaching Assistant. Your overview might literally be the definition of a teaching assistant.
Monitor. Proctor. Or, more specifically, Lab Assistant or Lab TA.