Perhaps I am not being clear.
You send an email to the ticket system. Lets say your address is email@example.com
This creates a user. The email address for that user is now firstname.lastname@example.org. Since Admin panel -> Tickets -> Autoresponder: New Ticket is checked it tells the user (email@example.com) that they have a new ticket.
Since you have an Agent account (also firstname.lastname@example.org) and you have Admin panel -> Tickets -> Alerts and Notices: New Ticket Alert (Admin and Department Manager) checked if someone opens a ticket to a department that you are the manager for, or if your admin email is email@example.com then you will get a notice.
It sounds to me like this is working as expected. So unless you have more information (such as you are not the department manager or site admin) then this is exactly how osTicket is supposed to work.