RE: [exim] not showing domainname

> > How to make the headers of the messages transfered by Exim _not_ to
> > mention domainname?
>
> Exim is designed at a fundamental level to always use full Internet email
> addresses. You cannot do what you want with Exim.

Sure it can be done. Envelope time rules (Exim accepts message envelopes without domains)

recipient_unqualified_hosts = 127.0.0.1
sender_unqualified_hosts = 127.0.0.1

And if you are really wanting to remove domains from headers, you probably need to do a transport filter. Check Exim documentation for it.