Country is proper but not for the reason you assert. Let's look at the definition of the word;Thanks for your response and, finally, agreement.
A country is a region that is identified as a distinct entity in political geography.
What defines a country?
Tract of land inhabited by people of one or several nations. Whereas the term 'country' emphasizes the physical dimensions and boundaries of a geographical area, 'nation' emphasizes a particular community of people with shared history and culture, and 'state' a self-governing legal and political entity.
The geographical area of the 13 Free and Independent States, united in a Confederation was what was meant by the use of the word 'country', the plural -'countries' in the treaty refers to both of the parties to the treaty.
In reference to the 'United States' it recognized the distinct entity that was the Confederation.
In reference to 'Great Britain' it recognized the distinct entity that is the Kingdom.
It does not mean that 'country' means 'nation'. Each and every State in the Confederation is a seperate nation, by definition.
Yes, words do have meaning. Problem is, meanings are construed to give credence to one's point of view, incorrect as they are.