> The website/app instance is running on the phone/device/whatever. It is not passive. The user opened it ( or opted to keep it open in background ).

That's not how it works. Unless it's the foreground application, the application is not running; it's not kept open in the background. Instead, what happens is that the application registers with a push notification system when the user first opens it and logs in. When the user should be notified, the website sends a message to the push notification system's central servers, which notifies the phone (the push notification system does stay open in the background, but it's shared by all applications in the phone), which then starts the application on the phone to show the notification.

Thank you for this. It always helps to understand something better.

That said, does it change twitter's status as a website that just happens to have functionality X ( push notifications )?

Can WebNotify (or whatever) work? That thing that some sites nag me with could have a purpose? Does it work on mobile?

A user gets notifications without having to look at their phone. That seems passive? Sure, prior installation & configuration of the app days/months/years beforehand was an active step but once you give it permission to send you notifications all of those notifications are received without user intervention required.