Your insurance most likely did get your neighbor’s insurance to pay. It usually goes on behind the scenes and you would never know. As long as they had the address or name of where it occurred that’s all they would need. Insurance never pays if they can get someone else to! DD once hurt her arm falling at a movie theater and insurance asked where it occurred and I told them the name of the theater. I’m sure they filed with their insurance. I was injured at a sports stadium and same thing. I did get a follow up call from insurance asking for more details and I’m sure it was for information to file with their insurance. On my end my health insurance paid for everything but there is ultimately no way to know if they actually paid or if the property owners insurance paid.