It is informative that they only list two usages of the word with that meaning prior to the 1770s, and quite a lot more usages of the word to mean "townfolk."
I know Shakespeare and Blackstone both used the word exclusively in the context of Town Folk, and the King James bible mostly uses it in that manner, and the one or two occasions in which it seems to mean the word as citizens of a country could be attributable to Wycliffe's influence. Apparently he was very instrumental in spreading the bible to the Masses back around 1380 or so.
I still maintain that post 1760s usage of the word, and especially the American usage of the word is directly attributable to Vattel's usage of it in his Law of Nation's book. Again I point out that the normal English word to describe members of a nation was "Subjects."
They must have chosen to use the Swiss meaning, which even your Oxford Dictionary indicates was the less common usage, over the ubiquitous and familiar word "Subject", because of Vattel's influence.
I've examined four other English dictionaries of the time period, and none of them have the Swiss meaning in them at all.