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

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That feels like how Apple, Google and Samsung would answer this question so that they can sell their own infrastructure as a "Push". :)

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.