exim4 some config error causing error how to pinpoint

[Harry Putnam <reader@newsguy.com>]

I've attempted to setup exim4 on a second debian OS, using the
working configuration from the older one.

Before getting too detailed I think I see something in output of my
debug technique that indicated somewhere my host is telling exim the
wrong host name.

(dv is host with working config )

I use this technique; sending a mail like this:
mailx -v -s "TEST 160912_211756 dv" reader@newsguy.com < ~/txtmsg.txt
                               
I run that command then watch the smtp output from the little
test script named `tmail'.

On the working host the first bit of output:

 $ tmail reader@newsguy.com
sending like this:
mailx -v -s "TEST 160912_211756 dv" reader@newsguy.com < ~/txtmsg.txt
LOG: MAIN
  <= harry@dv U=harry P=local S=546

[...]

Now the second, non-working host:

 tmail reader@newsguy.com
sending like this: (Non-working host is `d2')
mailx -v -s "TEST 160912_211617 d2" reader@newsguy.com < ~/txtmsg.txt
                           
LOG: MAIN      
  <= harry@d2local.lan U=harry P=local S=440
 ^^^^^^^^^^^^^^^^^^^^^
Notice the above difference .. in the working host it shows
  harry@dv (hostname is dv.local.lan)
  In the non-working host:
  harry@d2local.lan (hostname is d2.local.lan)

Somewhere in my setup exim4 is finding the hostname without the
separating dot in d2.local.lan   and is using d2local.lan

Which may be causing the failure:  Following the smtp otuput:

-------       -------       ---=---       -------       -------

 delivering 1bjcKi-0001Ey-9B
R: nonlocal for reader@newsguy.com
LOG: MAIN
  ** reader@newsguy.com R=nonlocal: Mailing to remote domains not supported
LOG: MAIN
  <= <> R=1bjcKi-0001Ey-9B U=Debian-exim P=local S=1650
LOG: MAIN
  Completed
delivering 1bjcKi-0001F0-Fy
R: nonlocal for harry@d2local.lan
LOG: MAIN
  ** harry@d2local.lan R=nonlocal: Mailing to remote domains not supported
LOG: MAIN
  Frozen (delivery error message)

------- ------- ---=--- ------- -------

Or am I making too much of the missing dot and it really is that I've
missed an important setting that lets exim4 deliver to outside of lan?