[Exim] local_domains behind router - the answer

I searched throuth FAQ, mailing list archives, and a kazillion websites and
could not figure out why I could not send from my workstation to the
server -- I am behind a router.

During the Linux install process, the /etc/hosts file was set to my internal
IP, as part of the network setup. The solution was to change it to the
public IP, without touching anything else.

I hope this helps someone else. It took me 3 days to figure it out... and
I've been working on systems for years.