Thanx for the screenshots!
Just for my own mental organization and verification of my understanding of your situation: Toggles are remaining on in both the Event Recording Smart AI Detections setting page and the Notifications Setting Page.
However, when you close the app you don’t receive any push notification alerts for Person Detections on your phone.
Other than the setting I mentioned above, I’m not sure on how to remedy this as I am NOT an iOS user.
The first screenshot you posted, however, is of interest and is an additional issue you may be experiencing:
What you have circled is the Quick Filter Person Event toggle from the Events Tab. It has been reported in the past that some Quick Filter toggles would revert to off when the iOS app is closed and would not hold their setting:
However it is just a filter toggle, and not a recording setting or notification setting toggle. It only changes which server event videos you see and which ones you hide. The videos you hide are still recorded on the server and should still be sending you push notifications.
If you toggle it back on, or use the funnel button to enable the Person Filter at the bottom (checked), you should then see all person tagged AI videos during the time the app is closed. It should NOT, however, affect you getting your push notifications.
What is your iOS App Version?