Accident caused, damage reported – thing with it off the table?

Hello, I have recently built a self-caused accident, it is my first one, so I don’t know how to deal with the insurance. It’s probably a minor damage (less than 700 euros from the accident opponent) while parking out, waited for the victim and then called the insurance, described the accident and sent pictures of the damage by mail as requested by the employee. Is this done for me? I read on the Internet that a phone damage report is not enough and still a written must take place until the insurance is active. Do I have to become active myself? or do I maybe get mail from the insurance with questions about the approach? The car is also not insured on me, if this should play a role. Would I be happy if you bring some light into the dark. Insurance is the HUK.