State has the same meaning today as it did during the 18th century.
STATE (Dictionary of the English language by Samuel Johnson (1768, 3rd edition) & 1792 edition) [...] 5. The community; the public; the commonwealth. 6. A republick; a government not monarchical. [...] 13. The principal perſon in the government.
STATE (Merriam-Webster Online) 5a. : a politically organized body of people usually occupying a definite territory; especially : one that is sovereign b. : the political organization of such a body of people c. : a government or politically organized society having a particular character <a police state> <the welfare state> 6 : the operations or concerns of the government of a country 7a : one of the constituent units of a nation having a federal government <the fifty states> b plural capitalized : The United States of America 8 : the territory of a state
The 18th Century (18th) definitions and the Merriam-Webster (MW) definitions both have the same meanings, but the word state itself has multiple meanings. It can mean the public (people), the nation/government (commonwealth), or the physical land division/territory of a nation.
Province has a similar meaning:
[PROVINCE] (18th) 1. A conquered country; a country governed by a delegate [...] 3. A region; a tract.
[PROVINCE] (MW) 1.a : a country or region brought under the control of the ancient Roman government b : an administrative district or division of a country
The USA began as colonies of the British
The strip of land along the eastern seacoast was settled primarily by English colonists in the 17th century, along with much smaller numbers of Dutch and Swedes. [...]
Each of the 13 American colonies had a slightly different governmental structure. Typically a colony was ruled by a governor appointed from London who controlled the executive administration and relied upon a locally elected legislature to vote taxes and make laws.
Today, the USA is a collection of different political entities with individual state laws/constitutions, that share sovereignty with the federal government which has specific delegated powers. How well this matches with particular founding fathers visions Jefferson/Hamilton of a weak/strong central government could fill a book (and is a separate question).
A state of the United States of America is one of the fifty constituent political entities that shares its sovereignty with the United States federal government. Because of the shared sovereignty between each U.S. state and the U.S. federal government, an American is a citizen of both the federal republic and of his or her state of domicile.