I'm posting my solution so it will end up in the archives.

This setup only SENDS mail, like I wanted.

Ran `eximconfig` and selected option #2 "Internet site using smarthost".
Accepted default (hostname) for 'visible' mail name.
Selected 'none' for: "other domains", "relay for other domains", and "relay
for other networks".
Set smarthost to 'smtp.west.cox.net'.
Redirected root & postmaster to a user account.
Accepted default for "/etc/aliases file".

After `eximconfig` completed, I edited the /etc/exim/exim.conf file. I set
'primary_hostname' to an empty string and changed 'qualify_domain' to
"cox.net".

  primary_hostname =
  qualify_domain = cox.net

I can now send e-mail to the internet, but I still had a "reply to"
addressing problem. To fix that I added to .muttrc:
  my_hdr From: Bart Garst <bartgarst@cox.net>

Thanks to Kurt & Jeremy for your help.

Bart