I hate to be pedantic but these are not push notifications. Those are the ones that can be sent even if the webapp is closed, for example, this Chrome demo: https://developers.google.com/web/fundamentals/getting-start...
WebPush is (going to be) an IETF standard: https://datatracker.ietf.org/wg/webpush/documents/
> Pushed responses are always associated with an explicit request from the client.
I'd rather see additional frame types defined that allow for bidirectional communication (i.e. websockets 2.0) than overriding and conflating the PUSH_PROMISE frame type.