Start-up problems – Helping technicians

Hello, I’m not so gifted technically, so I need some help. I bought a 2.5-year-old Peugeot Satelis, he ran perfectly during the test drive, the next day the seller picked up the ignition during the process – after that the scooter had start problems, but the battery symbol flashed. Somehow he found himself then and ran. After about 50 KM drive I had to fill up and he jumped on tadelos afterwards, after that he had 2 days later I wanted to start it in the morning, the starter turned it but it didn’t have it together. I then removed the battery and charged it. After charging I left you standing 24 hours without electricity, before installing the charger hung on it and after 2 minutes the charger showed fully, about 13V. Built in, scooter started, driven! 2 days later I wanted to take it out on the track twice and started it (it was an hour each) and it started again. I got a new battery, charged and built into it. E-starter pressed and again it just rolled. After a moment only ignition it then jumped on. I then drove over 100 KM – while driving it does not cause any problems. Battery symbol does not glow, engine does not go out, runs round. From my point of view only the battery can have been the problem, I have of course concerns that you drive somewhere and then it jumps suddenly n Would someone tell me what else to check? Would the LiMa be defective during the ride? Would the starter not be defective at the beginning? Would the petrol pump be defective during the ride? Thank you for a few tips!