I find it very interesting, and disturbing, that the UN refers to its members as "states" and not "nations" as in United Nations................
"State" is not an unheard of term for an autonomous sovereign nation. It doesn't actually mean the way it is used in the US, at least since federalism is nowhere to be bound. It gives a little insight into what the founders intended (and what we got now ain't it).
In international law, "state" is probably the correct term for a sovereign governmental entity. Nation is more of a cultural term. Thus one "state" could have many "nations" in it in a multi-ethnic empire - for example the old Soviet Union, Czechoslovakia or Yugoslavia. You might even argue Iraq and the United Kingdom are that way right now. Similarly, you could have one "Nation" split into multiple "states" like East and West Germany. The trend in the past century has been for the "states" and "nations" align.
We run into the problem of using the term "state" in English to also mean political subdivision in addition to sovereign government. There is no good English word meaning sovereign government and nothing else. State, nation and country all fall short.
The term state is normally used to refer to a sovereign geographical and political entity.
It's our interpretation of the word state to mean a relatively weak political entity under a strong national government that is incorrect. Our government was formed as a group of states that joined together in a federation to free themselves from British rule.
The UN isn't using the term state incorrectly. We've forgotten what states are.