Good news I got from the Bus Company today. I will not have to pay the fine any more.
The letter was a bit confusing to me. First it said the bus controller had handled the case correctly, and that all "Schwarzfahrer" are treated the same way, whether they intentionally or unintentionally violate the law.
Then it said in my case it was an error and therefore they will remove the fine. This case, however, is treated as an exception for only once and I should not tell anybody about it (so you guys, please don't tell anyone)