After research, I found a solution for my problem. Before I joined the exams, I always made sure that the camera settings match the requirements (for example, that the screen of my laptop is clearly visible in the camera, or that my workplace is visible).
Unfortunately, I did not close my camera before joining the meeting. I found out that the camera can only work with one application at a time. Of course it cannot work on Zoom, if I have already opened it.
I can recommend to everyone to make sure that the camera is not running (also make sure to close apps that use the camera), before joining a Zoom meeting where the camera is required.
I found out that by default, Zoom lets attendees change their own names during a meeting at any point. However, sometimes this option seems to be missing. The reason for this is that Zoom has a setting that lets Hosts choose whether they would like to allow the attendees of a meeting to be able to change their name once a meeting has started.
I found this information here: https://nerdschalk.com/why-cant-i-change-my-name-on-zoom/
There are several ways to solve this problem. The easiest way is probably to ask the meeting host to enable the setting, that allows participants to change the name. I solved the problem by leaving the meeting, logging into the Zoom app, changing the name there, and then rejoining the meeting. It would have been much easier to just ask the host to change this setting. Fortunately, I was able to rejoin the meeting fairly soon, so I didn't miss any part of the lecture.