Through past research, I've found that users connect better with the term 'notifications' rather than 'push notifications' (as push is based on quite a techy concept of the notification being pushed from a server).
However, a project I'm working on is intending to have the same notification content viewable via a push notification (outside app) and in the actual app as part of an activity feed and being told what has happened since last logged in. The reason these same notifications are intended to be in the in-app format as well is because users may have notifications turned off at the OS which is obviously uncontrollable from the app.
So how would it be best to distinguish the difference between these push notifications and in-app notifications?