Hey Todd,
Glad to know that you're loving Freshdesk. And I am sorry that I took way too long to sort out this annoyance for you.
Whenever a customer responds, you will receive a "New Reply Received" email which can be configured from Admin>>Email Notifications.
Also, the ticket is automatically moved to the Open status whenever the reply comes in and this is taken care by a default Observer rule called "Automatically reopen tickets when the customer responds".
Head over to Admin>>Observer and click on "Edit" next to this rule. This also sends an email "Ticket reopened" to the agent.
You might have to remove this action alone or turn off the email notification.
Let me know if this helped, Todd.
Anna
Thanks Anna. This was resolved by chatting with your support. It was an observer setting as you mention.
Have a great day/night!