The usual translation of Medinat Yisrael into English is "State of Israel." That seems strange if one thinks of a "state" like Florida within the United States of America. But that's not the meaning of "state" in Israel's case. Rather, "state" is the equivalent to "country."
The United Nations officially has "member states" (the USA, Israel, etc.). But the term could just as easily be "member countries" or "member nations."
Your confusion comes from the fact that in the United States, we have mangled the meaning of the word state. In the rest of the world, since time immemorial, the word STATE had the same meaning as that which we now apply to COUNTRY. When the thirteen colonies declared independence, they said that the thirteen colonies were "free and indendent states". Then, eleven years later, the Constitution was writeen which federated the states together into a single nation, while retaining their independent statehood.
Since then, the independence of the states has become less and less, and we now think of the states as creations of the federal government, and not the other way around (and to be fair, except for the original thirteen and Texas, they are). In our minds, we now think of a state as a regional political authority, and that definition is creeping throughout the world, but in general usage, state can still mean "independent nation".
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