EU is definitely not a synonym to Europe. That would exclude UK, Norway and other countries that are certainly in Europe. I've never heard anyone even thinking those two things are the same.
The citizens of St. Barts are French. Are you claiming they aren't European? Tell me, do they become European when they visit the Metropole? Does this quality vanish when they return to the Caribbean? I'd love an explanation!
This entire thread is a response to a claim about Europeans, and I took pains to draw out that distinction. Yet here you are talking about Europe. Why?
People born there? French kids of French parents.
If St Barth had different demographics, there might be a more meaningful debate to be had about this. It’s not exactly the melting pot you might expect from an old colony.