push and local notifications. What is the fundamental difference and how to implement it in full screen?

Question

Android

I continue to delve into nativescript.
Tightened the local-notifications plugin. Hung up on the pusher channel. Everything works - a notification pops up on top of the phone.
But here I heard more about push notifications. But for some reason, only through firebase is the implementation. I think it will be too much for me.
Let's say the app is running in the background. I want to receive a notification - a full-screen pop-up window with buttons to go to the application.
Is this a splash screen? where to look?