Myth
"Convention of States" is just a term invented by the Convention of States organization. The convention in Article V is not a convention of states.
Fact
The term "convention of states" is not a recently invented phrase. It was widely used during the founding era. The founders even referred to the Philadelphia convention as a convention of the states.
Fact
The Article V convention IS a convention of states.
How do we know?
We have the Supreme Court and we have legislative action from the states both referring to the Article V convention as a convention of the states.
Smith v Union Bank (1831)
The Supreme Court specifically refers to the Article V convention as a convention of the states...
Whether it would or would not be politic to establish a different rule by a convention of the states under constitutional sanction is not a question for our consideration.
source: https://supreme.justia.com/cases/federal/us/30/518/
Virginia General Assembly
The very first Article V application, submitted by Virginia states...
“Happily for their wishes, the Constitution hath presented an alternative, by admitting the submission to a Convention of the States”
source: https://www.loc.gov/resource/rbpe.1790100c/
Fun Fact: the Bill of Rights
This was Virginia gearing up for a convention to propose the Bill of Rights should Congress have refused to act.
What if Congress never did act and we had no recourse through a convention? There would be no way to force Congress’s hand. We would just have to keep electing better people and hope that at some point, Congress would decide to limit its own power. You could easily argue that it was the threat of a convention that forced Congress to act, because the convention is an institutional power rival to Congress.
We are left to wonder: Would we even have a Bill of Rights today if the states didn’t have the authority to begin the amendment process themselves?