The trap includes the item name. It doesn't and wouldn't include the parent device name as that is not what the event is against.
Might also want to try the EventManager trap format instead of the nhLiveAlarm type as it may provide more useful information.
The Notification email sent is configured to provide the parent device name in the subject. Check Administration->Configuration Settings->Notifications to find and edit the one sending that email to see how it's set up.