I think the root cause of this is that your incident task process and request task processes work in different ways. My guess is that the incident task process has an assignment and/or a reminder at the beginning of the task process and this sends the message out. Typically this mechanism could be used for all task processes in all domains, but you may well find it is not set up that way for request (and change) task processes.
So compare your task process for incident request and see how they are different, especially right at the start.
As an FYI our OOTB processes have the various domain tasks processes set up the same way to avoid this issue. They also send a notification back to the assignee of the main process when the task is completed so the assignee of the incident or request gets to know someone has completed a task.
Your last set of windows shows a slight confusing between task assignments and process assignments. The first window is the assignment for the request, the second is for the task. Two different objects and hence two different windows with different contents.
You might also consider never using assignments to send messages, but always using reminders. We tend to do this so customers have the same spot to edit when changing messages instead of pecking around in both assignments and reminders. Both methods work, but you need to be clear about which one you are using. Just a thought.