So I would expect the phrase "US government" to include the government of Washington State, along with all other governments within the US. Do you really use it only to describe the federal government?
Yes. Pretty much, "US government" refers to the Feds.
What do you say instead when you mean the federal government *and* the government of the States and other territories collectively?
We hardly ever talk about that. At levels lower than the Feds, we talk about "State and local governments" from time to time.
But the States are only subordinate to the Feds on Constitutional matters (there are things only allowed to the Feds, and things only forbidden to the Feds), so much of State law is completely orthogonal to Federal law. This is generally not true at State level (cities and towns can pass their own laws with the permission of the State, but only with permission).
But at the State level, that's just not true (an example: Murder is a State-level crime. It's only a Federal crime if it takes place in an area not under jurisdiction of State law (military reservation, for example, which is not subject to State law, even if it sits entirely within a single State)).
So we very seldom clump all law-making bodies from the Feds on down together.