That's simply another pedantic distinction to justify state-owned and state-mandated organizations not providing core government services.

These terms usually mean the same thing. Do you want to explain the difference you're trying to highlight?

On this side of the atlantic at least, the government tends to just mean the prime minister / head of government and their cabinet (and to an extent the associated political parties in charge).

Ok so what is a general term for your courts, your legislature, your bodies responsible for defense and foreign policy, your securities and banking, your monetary policy, etc? It's all government.

The state? Government just isn't used as a term for that here, except as an encroaching americanism.

In amareicanese the government is called the administration.