"Indigenous American and Hindu culture, for example, have traditionally had no such division between religion and culture."

Isn't that just a bias from watching too much discovery channel? We are used to seeing other cultures presented as being preoccupied with fancy rain dances, perhaps because they look good on television. But they must have a "normal day routine", too, which I would consider to be culture as well? It might be the "bag of rice in china phenomenon" - mundane daily life might not make for interesting movies, so we disproportionally often get to see special events from other countries.