Yes all clients have push notifications until they are closed. Apple/Google just maintains another connection to keep the push notifications alive for when the app is not open.
Discussion
Just to make sure i wasn’t taking crazy pills i put our argument into 4o to see what it thought:
https://chatgpt.com/share/67fd79aa-f54c-800f-b755-ec6b5787f60c
That feels like how Apple, Google and Samsung would answer this question so that they can sell their own infrastructure as a "Push". :)
have to agree with nostr:nprofile1qqsr9cvzwc652r4m83d86ykplrnm9dg5gwdvzzn8ameanlvut35wy3gpzpmhxue69uhkummnw3ezuamfdejsz9thwden5te0wfjkccte9ekk7um5wgh8qatzqyt8wumn8ghj7un9d3shjtnswf5k6ctv9ehx2aq38qmrf on this one
An interesting history: BlackBerry, the creator of Push Notifications, did sell the BlackBerry Enterprise Server that would run inside the NAT and open connections directly to phones. Outside of enterprise service, the phone has to make the connection to their server as it happens today.
first pull, then subsequently push (and when network changes, repeat).
this is a mouthful though, so they just say the one that sounds cooler
This is like suggesting a laptop is named wrong because most of the time they're not placed on people's actual laps.