TIL there’s actually a difference between coffin and casket. Coffins are tapered to the body (6 or 8 sides), usually with a separate lid, typically far cheaper. A casket is a rectangle with a hinged lid, typically with rails attached for carrying and usually far more ornate & expensive.
Everyone in my family does this. They send you the ashes when they’re done. I’m not sure of the price, but it’s trivial. Shipping and handling mostly.