At last it runs. But it was a journey. I used my laptop instead of my PC because it has an USB-C port and runs a Linux distribution. And I bought a new USB-C to USB-C cable. But now the funny thing: This one doesn’t worked! I always get the same failure message as this guy here: Phone stuck in fastboot mode, unable to install Fairphone OS
So I used my old USB 3.0 to USB-C cable and this one worked.
Fairphone guys, you have to fix this!