"American" would be ambiguous in this context, right? Both the Cherokee and the English-speaking residents of the USA are American, but the specific point here was about whether Sequoyah was only a Cherokee speaker or if he had any knowledge of English, Spanish, French, Latin or any other European language. In this context, saying that Sequoyah's father was "American" would not make any point - it wouldn't tell anyone reading the comment that he was not Cherokee nor from any other native population; whereas European makes that point succinctly.

I find it interesting that you think "served in the U.S. Army" isn't American enough for you.

Foreigners can serve in the US Army. Native Americans weren't automatically US citizens until 1924, but were considered citizens of their sovereign tribe.

European here clearly means both "from Europe" (eg, Latin, Greek, and Cyrillic letters are European, not American), as well as "European Americans" (ie, Americans of European ancestry, and often with cultural ties to Europe.) Just like how "Asian" doesn't always mean "born in Asia", or how "Anglo" can refer to non-Hispanic white Americans rather than being specifically related to England.

Trading with the Spanish in Florida, English ships, or French trappers would all count as "contact with Europeans", and not simply "Americans".

Finally, recall that at the time "American" was a state of mind. A Loyalist at the time would not consider themselves "American", and a Patriot considered a Loyalist to be "inimical to the liberties of America". How do you know if Sequoyah’s father was an American or a Loyalist?