I had always thought that Luke was a Greek physician who lived in Antioch before he fell in with this new bunch who called themselves Christian.
You are correct in that the "Greek Physician" identification of Luke is the prevailing view.
I have read various and conflicting accounts of Luke's identification and, frankly, it seems there is no solid answer.
I will not argue with your understanding except to note none of us really knows.
I hereby retract my claim. :-)