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
account1@example.com to
account2@example.com are being delivered.
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Lessons learned:
server1.example.com DOES NOT equal example.com
mydestination is vital.
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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?