Disable exim4 logging

You seems to be sending emails from your webserver user account (www-data) through /usr/sbin/sendmail interface but your system doesn't accept e-mails for this address (www-data@mydomain.com) so this address fails 'sender verification' on receiving server.

You have two options. First - enable www-data account to receive messages from external systems. Second - change envelope sender address from www-data to something different by using -f option for /usr/sbin/sendmail.

As you are trying to send the messages to yourself, configure Exim to authenticate with your gmail credentials when sending the messages. This should work for low volumes that are not being forwarded.

Add your credentials to /etc/exim/passwd.client. You should secure this file so that Exim can read it but not general users. I would not recommend this for high volumes of email or for email forwarding to other users.