Question to linguistic pedants out there; would that be "the"?

That would be the article, not the name.

To linguistic pedants out there: would that also mean that "united" is not a name?

Yes, united in "the united states" is an adjective. States is a plural name.

anyway, my reply was to "See the first word in the name of the country."

The first word in the name of the country is "the" and the first name in the name of the country is "states". Right?

yeah, it's not clear to me whether the proper way to call the United States is the United States or United states.

before the Civil War it was often "these United States"

Do you mean noun?

See the second word.