Love this, glad we are focusing more globally.
When I lived abroad, I learned to introduce myself as being from the Southeast USA. Broad and also specific enough to give people an out if they just wanted to know my country, and an in if they knew about our legendary Cajun food
American is a misnomer as anyone from North and South America is an American technically. My Latino friends would get a bit irritated when I introduced myself as American out of habit.
Also, LA is very contextual. You have to listen to their accent to infer whether it's Louisiana or Los Angeles they're referring to.