This is correct, but my understanding of it is that the push notification (which is not the same thing as the actual "Notification" that is shown on the screen) basically contains a "hey $DEVICE, go talk to $APP_NOTO server they got something for you".
APNS just taps on the device's metaphorical shoulder and hands them a courtesy phone "call for you sir"
APNS just taps on the device's metaphorical shoulder and hands them a courtesy phone "call for you sir"