A REST Push notification is like an event. Instead of you polling the server (issuing the HTTP commands) every few seconds in order to get the state of the alarm system (i.e. armed away, ready state, armed stay, alarm sounding, etc), REST technology provides the ability to "push" events out to the connected clients. Therefore you would get notifications (almost) instantly instead of there being a lag.
Having a program poll the website on the PI is an expensive time/consuming operation. It chews up a lot of resources and sometimes can overwhelm the webserver itself.
I know the documentation for the command set is on the website. However, REST developers usually provide push notifications for event handling. I wasn't sure if there were some undocumented features that weren't listed.
I hope I am explaining myself okay. Sometimes I don't make sense -- or at least that's what my wife tells me
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Thanks.