Earlier this year, I documented one of the reasons we know an Article V convention is a “convention of the states” rather than a mass popular gathering: Founding Era documents tell us so. I listed several such documents. (Subsequent to the Founding, in the case of Smith v. Union Bank, the Supreme Court also referred to an Article V convention as a “convention of the states.”)
Here is another piece of evidence:
In 1788, New York ratified the Constitution, but the state ratifying convention called for extensive amendment. In a circular letter to the other states, it urged that an amendments convention be called. On February 4, 1789, therefore, the New York state assembly, the lower house of the legislature, debated whether to submit an application to Congress for an Article V convention. Only four assemblymen spoke to the issue, but two of them characterized the gathering as a “convention of the states.”
Both of those two were highly significant figures. Samuel Jones had been a key member of the ratifying convention. So also had John Lansing, Jr., a respected judge. Lansing, furthermore, had been a delegate to the Constitutional Convention itself.
No one called the amendments convention anything else, although the eventual application used the term “Convention of Deputies from the several States.” The term “Deputy” was a synonym for “agent”—in this case, the agent of one’s state.
You can read the legislative proceedings in volume 23 of the Documentary History of the Ratification of the United States Constitution.
I have updated the earlier posting accordingly.