Congratulations! I am surprised but will try to understand better how that mattered. How did you figure it out?
After installing the new Zener diode, I measured again and then noticed that 0V is still displayed. On the other board it was about 0.7V. So I looked for the only difference: The capacitor C3. After unsoldering the capacitor, the values of the diode fit and after installing another capacitor everything worked as it should. I suspect that a lowESR has special characteristics that cause problems in the 5V region.