I know this is an old ticket but I'm experiencing the same thing. I've tested it just about everyway possible. I sent an email from an email that was a client email, from a non-client email, from the web portal while logged in and not logged in and all go to status "open" and don't notify my while logged in. The only way is to click on the "ticket(s) awaiting reply" link and then change the dropdown on the left hand side to "any" to see all the tickets that I submitted.
Any fix for this? it used to work fine that any emails that came in, or customers replied to would show up as a notification and in the list until I closed them or moved them to another status.