Louisiana has a very unique history (exploration, occupation, settlement) - and it's history is reflected in it's food.
Rustic country cooking is usually attributed to the Acadian French who migrated down from Canada to Louisiana - the Cajuns. The most prevalent "city" style of cooking in New Orleans is Creole - a blend of European French, Spanish, African - with some Cajun and Native American influences (guess there is a minor touch of British, too) ... to one degree or another. I don't think many Creole dishes can be linked to just one source/origin.
Now, before someone jumps on me for saying Cajun cooking is rustinc country cooking ... I never said it was unsophisticated. Look at Chef Paul Prudhomme's family's cookbook. The ingredients may be simple - properly executing the dish can be somewhat of a challenge.
I will agree that jambalaya is a dish where the rice is cooked in it, although I have eaten in a couple of places where it was cooked and served separately (served on top of the rice) - one place was in Jacksonville, FL and I think the other was Houston, TX. Gumbo can range from a soup to a stew ... usually served over rice .. but I have had it served as a soup without rice or with the rice as a side dish.
We could have a similar debate over the origin and authentic ingredients in Brunswick Stew.