Thanks Karl, today I did the following tests:
1. I had disconnected starter motor from relay and pressed few times the button: 3 times the relay had clicked, then has become quiet, the fuse blown.
2. I had removed the bulb from back light (the starter still disconnected) and pressed few times the button: 3 times the relay had clicked, then has become quiet, the fuse blown. HERE I NOTICED THAT THE BACK LIGHT IS DAMAGED ON ITS RIGHT SIDE, IT REMINDS ME THAT ONE DAY I FOUND SOME SAND ON THE SIDE OF MY SCOOTER ON AN EXTERNAL PARKING, IT SEEMED LIKE SOMEBODY PUSHED IT (BY MISTAKE WHEN BACKING WITH A CAR), THE SCOOTER FELL DOWN AND THEN THE OFFENDER PUT IT UP AND LEFT.
How I understand it: Starter is innocent, some wiring between hadlebar and back light makes short circuit (or also the relay might do it?). I cannot see any "suspicious" wire there for now..