Hello Chris
BGP will choose the router ID based first on if the router-id command is present. It is not in this case. Secondly, it will choose the highest IPv4 IP address of a loopback interface. You are correct, that both R1 and R3 have the same loopback address (seen in the configs), and thus would be advertising the same router-id. So the next step would be to determine the lowest neighbour IP address, which is indeed 192.168.12.1. So you are correct, it does prefer 192.168.12.1 but not because of the router ID but because of the lowest neighbour IP address. I’ll inform @ReneMolenaar about the correction.
Thanks!
Laz