help with FQDN on lan

My domain name is registered and DNS for it is hosted by a third party
(easydns.com). My public zone file contains only a listing for the
domain name and two hosts (eg mydomain.com, www.mydomain.com and
mail.mydomain.com) and all three point to my 1 public ip address which
is connected by a router to my lan (address 192.168.1.0). The router
forwards ports 80 to my internal webserver and 25 to my internal mail
server.

Mail and web services work fine.

Inside my lan, I am running my own DNS server on Redhat 8.0 Bind version
9.2.1 with unchanged configuration and zone files to resolve queries
from my hosts on my lan which includes linux and mswindows computers.
Public name resolution works fine (eg www.google.com) but I can't figure
out how to configure named to resolve hosts on my lan. This is not a
major deal because I have been using the hosts file just fine, but I
would like to use BIND for the usual reason of having to avoid updating
20 hosts' hosts files.

Several interesting issues have come up while doing this.
Although telnet and smbclient resolve using the hosts file, the program
host does not. It gives me an error
host fred not found: error 3(NXDOMAIN)
Why doe the host program resolve differently?

Here is host.conf
oder hosts,bind
multi on

Here is /etc/hosts
127.0.0.1 localhost.localdomain localhost
192.168.1.101 mailserver.mydomain.com mailserver
192.168.1.102 webserver.mydomain.com webserver
192.168.1.103 fred.mydomain.com fred

I would also like to know the proper way to name hosts on my private
lan. Should they only be named with unqualified names like mailserver
and webserver and joe and fred or should they have FQDN like
mailserver.mydomain.com, webserver.mydomain.com, joe.mydomain.com,
fred.mydomain.com?

If I try to resolve fred.mydomain.com the queries will be directed to
the easydns server which is authoritative for mydomain.com and not read
my local zone file. I don't want to run an unregistered domain on my lan
but i supose I could. I really would like to keep my computers part of
my registered domain but I can't figure out how. Please help.