Gmail is adding the Outbound Mail within the Inbox
Summary
If you are seeing that Outbound notifications are being received as Inbound Mail it could be that you have a Mail Loop within your system or that you have a Gmail Message filter that is adding the outbound sent mail into the Inbox of the mail server which is causing the mail to be processed as incoming mail.
This is undesirable as it consumes your Plan allowance for no useful outcome and would be in your interest to diagnose and limit such traffic.
Mail Loops
The typical cause of mail loops is that there is a recipient that is an incoming Mail server or is an alias of an Incoming Mail Server.
For further info see the following pages:
JEMHC documentation page: Prevent Mail Loops
JEMH documentation page: How to prevent Mail Loops
Gmail Message Filter
If the above does not apply and you are using the same Mail Server account for the Outbound and Inbound connections, then it may be that there is a filter that is configured to prevent mail from a specific domain from going to spam.
For more info see the following public forum: How can I stop sent items from appearing in my inbox?
Based on the above forum it appears when such a filter exists and matches it will result in the Sent mail also being added to the Inbox of the Mail Server which will then be treated as inbound Mail and will be processed by JEMH.
Below is an example of such a filter:
Solutions for this issue
Remove this filter from the Mail Server.
Modify the Filter so that it does not match the Email Address that is used for Outbound Notifications as this means that the filter would not add the emails sent from this address within the Inbox. e.g.
From: @thepluginpeople.com -outbound@thepluginpeople.com
Mail Server configurations is your responsible and is not within our support scope.