Yes, sms is "actually push" all the way down to the transport layer.

As far as I know, this is still what push notifications are built upon for an idle/sleeping device.

Carrier infrastructure knows which tower you last connected to, instructs that specific tower to broadcast a message telling your phone to wake up and fetch the remaining 80% of the notification content (the sms bit is usually just enough for your device to learn the UUIDs of the notifications)