What is a TA called when he/she is in charge of organization? I am a non-native speaker and writing my cv. In my subject (mathematics), there are small exercise groups where you have to grade the students' homework and discuss it with them; some are hold by students; some by PhD students or postdocs. 

What are the correct terms if you have done the following things:


You are a (PhD) student student holding the exercise group, grading homework, but nothing more.
You are the one who is in charge of organization, i.e., you design questions for homework and the exam, you make announcements, moderate the meeting of those people having an exercise group (see 1), but you don't have an own exercise group.
Same as 2, but you have an own exercise group.
The course is so small that there is only one exercise group. You are holding this group and are in charge of the organization.


Thank you very much!
First to answer your titular question:


  What is a TA called when he/she is in charge of organization?


Such a position does (in theory) not exist, as a TA as by definition supposed to "assist" somebody else and not run the show on her/his own (I am aware that this is not how it sometimes works in practice).

Generally, you need to keep in mind that title and scope of academic positions, especially regarding teaching, are quite different between US and Germany, so you will likely not be able to do an accurate 1-to-1 mapping. I would write "Wissenschaftlicher Mitarbeiter" in the CV and explain what you actually did. Translating it to english is prone to confusion.

If you write TA, people will underestimate your responsibility in the course, if you write something like lecturer, people may infer that you already had a faculty position and will be confused by the perceived mismatch with the rest of your CV.