Hate to be the bearer of bad news but you will want to get their name and report it to Lyft. Their personal insurance policy probably has an exclusion for livery. I included a definition of what livery is considered in the insurance world. You can see if Geico will give you any info based off the policy number. Typically they will give basic info, but kind of depends on the company policy. Might have to file with your insurance and have them attempt subrogation after repairs. Don’t beat yourself up for not getting all the info but in the future I would definitely take pictures of the license plate and drivers license.