I always listen to my parents’ kid.

It’s an official Sony thing.

The difference is just whether the client app gets notified of new messages/replies by asking the server “Is there anything new?” on a set schedule (pull), or if the server sends the notification “Here’s something new!” only when there is a new one (push).

Yep. I’m not sure if OP was looking for technically-pull clients specifically, though I have to imagine conventional notifications are a basic feature even on iOS.

Checking at a regular interval is “pull” (not push) though. Push means the client keeps a low-bandwidth connection to the server open so the server can notify the client as soon as something has happened.