First, let's assume you didn't modify the OOB Process record that gets used when you open a new Incident or update an existing Incident, so that the standard Processes are getting called and not a custom Process that may be calling your notification engine more than once.
Second - when something breaks, go back to the last time it was working. OOB, notifications are triggered correctly on IM Open and IM Update. If this has stopped working for you, what customizations have you made to the tool? When was the last time it was working correctly, and what did you change since then?
Third - please grab screenshots of the IM Open and IM Update notification records. Make sure that you include ALL notification records with those NAMES - not the 'IM Open_1' ID and 'IM Update_1' ID. If you are performing an in-place upgrade of HPSM, the upgrade creates new notification records with the same name and different IDs, but the add and update Processes call the notification definitions by name.
Fourth - check to see if any Macros have been created on the probsummary table that trigger notifications.
We'll start there, but, as always, YOU are the person most qualified in the world to determine what broke, because you know what you've done, and you're there to see what is in front of you. We'll advise as best as we can, but you're the person in the best position to solve this issue.