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?
I can only take an educated guess here, but the extra information that you are a PhD student can only be in your favour as it explains what you are doing with the other 50 % of your time and gives you additional good reason to reside in Germany. As this is a common situation (the vast majority of German PhD employments are like this), I doubt that anybody is going to bite your head off for telling the truth here.
That being said, the institution where you will do your PhD is likely to have some experts or other resources to assist you with this.