No - sorry - the host file 'trick' was not a solution for your problem and I apologize for sharing that.
The issue is really minor as the assumption is the Moodle is to be accessed from inside the private network as well as the public internet. The only way to be able to do that ... a FQDN that resolves both inside and outside to the correct IP address of the server (depending upon station making the request is coming from private network or public internet).
Actually, many who run apache servers have the FQDN hostname set in apache config - which then behaves pretty much like the restriction in Moodle.
One reason Moodle uses such restriction: the variable is used to construct links to images and files. The assumption, am guessing, is that moodle is to accessed from both inside and outside a private network and thus links need to be consistent. If Moodle was configured like apache, either the private IP, the public IP mapped to the private IP, or the internet domain (FQDN) or the internal LAN/WAN private network name could be used.
Let's say someone on the inside uses the private network name to build a bunch of links to files on the server. Those links would have:
http://someprivate.local/... blah ... in the URL.
A user hitting the Moodle from the outside (those links having been built from how the client accessed the server and now in the DB) and clicking on such a link would get 404 errors and correctly so ... someprivate.local cannot be found in outside DNS.
I see in http://www.synology.com/support/faq_show.php?q_id=404&lang=us
Also this:
http://forum.synology.com/wiki/index.php/Remote_Access_on_the_Synology_DiskStation
synology, BTW, looks like an impresive product ... wish I could afford such.
'spirit of sharing', Ken