Many thanks for your professional expertise and help.
This seems to have solved it.
Mails sent from Gmail arrived safely into their destination accounts.
Also mails sent from firstname.lastname@example.org
are being delivered.
server1.example.com DOES NOT equal example.com
mydestination is vital.
I've said in a post before, that one email account has the same name as an actual user on the server. Obviously the mail user would be much easier to modify. Should I do this?