Some people complain that they have too many notifications on wiki, where they fly-out can be overflowed. (A recent case in my mind about structured discussions.)
A way to solve that would be to provide a choice to have notifications on wiki displayed in the fly-out, and/or only in Special:Notifications. That option makes Special:Notifications as an inbox. It adds a third level of reading: alerts, notices, inbox.
Notifications in inbox should be counted as regular notifications, but could be visible at the bottom of the notifications fly-out: