I've only even seen one South Texas Cowboy tie his bandana like that. He might not be from around here, I've never had the chance to actually meet him.
If you tie your bandana like was done in the movie 'Appoloosa' it will pull off without choking you. You put the bandana around your neck with both ends hanging, tie a loose square knot on one side and put the other end through the knot. Slightly tighten the knot and it about as safe a way to wear a rag as there is. It does take just a bit of practice to get where the ends hang the same lenght but it's not that hard.