Okay, so...
We were now able to localize the problem. (thanks to Don & Gremlin

).
Specifically, there's a problem with the central electrical system. There's a loose connection on the socket for relay 103 (the preheating relay).
If the car doesn't preheat when starting, I briefly jiggle the relay. After that, everything works fine and it runs perfectly. Even the vibration from closing the door can trigger the loose connection.
At the next opportunity, the central electrical system will be disassembled and all connections will be checked.
By the way, the relay has been tested and is working perfectly.