What does commonwealth mean in US English?

As Wikipedia says:

A commonwealth is a traditional English term for a political community founded for the common good. Historically it has sometimes been synonymous with "republic". The noun "commonwealth", meaning "public welfare general good or advantage", dates from the 15th century.

The British Commonwealth (also known as The Commonwealth of Nations) is the biggest and most important commonwealth, so absent context indicating some other Commonwealth, anyone who says "the Commonwealth" is usually talking about the British Commonwealth.

The idea behind Commonwealth is in its etymology:

OED:

Etymology: In its history, like common weal n.; wealth , Middle English welthe , being a later formation, in same sense as weal, Old English wela . The two words were used indiscriminately [...]; but in the 16th cent. commonwealth became the ordinary English term in sense 2 [...].

In commonwealth, "common" = of the people; of the community, and "wealth (n.)" = a [large or entire] quantity.

Thus

2. The whole body of people constituting a nation or state, the body politic; a state, an independent community, esp. viewed as a body in which the whole people have a voice or an interest.

And thus

3.a. A state in which the supreme power is vested in the people; a republic or democratic state.

a1618 W. Raleigh Maxims of State (1651) 8 A Common-wealth is the swerving or depravation of a Free, or popular State, or the Government of the whole Multitude of the base and poorer Sort, without respect of the other Orders.

And thus as the states of America originated as autonomous entities:

3b. A state of the United States of America, esp. in the official titles of Kentucky, Massachusetts, Pennsylvania, and Virginia.

1779 Pennsylvania Gaz. 24 Mar. 4/3 All the best Whigs in the Commonwealth.

1870 J. R. Lowell Among my Bks. (1873) 1st Ser. 228 The sturdy commonwealths which have sprung from the seed of the Mayflower.

1967 Boston Herald 1 Apr. 3/1 The Commonwealth has been married to the Sales tax a year Saturday. Like many marriages, it hasn't lived altogether up to expectations but it's become essential anyway.

and, historically, the first "Commonwealth"

4.a. English History. The republican government established in England between the execution of Charles I in 1649 and the Restoration in 1660.

For the US states that use the term, it's a bit more specific than the ordinary meaning. Specifically, it means a representative democracy.

For people living in or near those states, we learn the specific meaning in grade school and then immediately forget it, so that it becomes just a meaningless word in the name of the state. There's no practical difference between being a commonwealth vs a state. (Or at least, I learned the meaning in grade school and then immediately forgot it.)

From virginia.gov:

Why is Virginia called a commonwealth? What other states are commonwealths?

According to the Hornbook of Virginia History, 4th ed., page 88:

"A commonwealth is 'a state in which the supreme power is vested in the people.' The term as an official designation was first used in Virginia during the Interregnum (1649-1660), the period between the reigns of Charles I and Charles II during which parliament's Oliver Cromwell as Lord Protector established a republican government known as the Commonwealth of England. Virginia became a royal colony again in 1660, and the word commonwealth was dropped from the governor's full title. When Virginia adopted its first constitution in 1776, the term commonwealth was reintroduced, most likely to emphasize that Virginia's new government was based upon the sovereignty of the people united for the common good, or common weal. The designation commonwealth of Virginia has been used in official records ever since. Three states beside Virginia adopted the appellation commonwealth: Kentucky, Massachusetts, and Pennsylvania."

From mass.gov:

Adams wrote: "There is, however, a peculiar sense in which the words republic, commonwealth, popular state, are used by English and French writers; who mean by them a democracy, or rather a representative democracy; a ‘government in one centre, and that centre the nation;’ that is to say, that centre a single assembly, chosen at stated periods by the people, and invested with the whole sovereignty, the whole legislative, executive, and judicial power, to be exercised in a body, or by committees, as they shall think proper." (Adams, John, and Charles Francis Adams. The Works of John Adams, Second President of the United States: with a Life of Author, Notes and Illustrations. Charles C. Little and James Brown, 1850-56, vol. 5, p. 454)

Despite the literal definitions, the etymology, and the example labelings (the former colonies associated with the UK, the 4 US states), like many political terms it doesn't mean what it says.

'Commonwealth' means whatever the labelers want it to be. At most it means 'some kind of government where we want people other than an aristocracy to have some sort of rights, and maybe some organization'. If there is any technical meaning, it gets it only from the very precise situation in which it is used. Though it sounds like there is some vague socialist ideal in it, hardly any examples of commonwealths seem to imply any kind of collective economies or coordinated planned markets.

In other words, you can't take the dictionary definition or any particular examples and then point at a new situation and say 'Oh yeah this is definitely a commonwealth' or 'Wow, no way is this a commonwealth because it violates this one single part of the definition.'. Hm, no, you're right, dictatorships tend not to be easily called commonwealths. But that's about it.

Etymonline says it best:

In the U.S., it forms a part of the official name of ...(irrelevant state names elided)... but has no special significance.

In other words, in the US, 'commonwealth' just means 'government of a particular entity synonymous with 'state'.