I have been offered a Ph.D. position with a TV-L13 (50%) salary. I am in the process of applying for an employment visa (as I am a non-EU citizen). The work contract provided by the university is for a part-time scientific associate. I am unsure of what exactly to write in the personal cover letter (explaining the exact purpose and duration of the stay) for the visa application. Should I write about both the part-time scientific associate position and Ph.D. or only about the part-time scientific associate position?
You should write both of them. You should make it clear that you pursue your education by doing PhD and in addition to that, you are supported by 50 percent tvl13 position. In addition to that you should attach the invitation letter to your docunents. In the letter they must clearly provide an approximation of your income. (You need this support to not put extra money in back and ease your way through the visa process, I'm not sure if 50% suffices).