What constitutes a tub? vs a shower? Is there a minimum curb height required?

There is no dimensional code definition for a bathtub or a shower.

ICC/ANSI A-117.1

The only dimensional requirements are the ones set by ADA which do not specify a height for shower curb or bathtub apron edge, but require a seat between 17 to 19 inches.

610.2 Bathtub Seats. The height of bathtub seats shall be 17 inches (430 mm) minimum and 19 inches (485 mm) maximum above the bathroom floor, measured to the top of the seat. Removable in-tub seats shall be 15 inches (380 mm) minimum and 16 inches (405 mm) maximum in depth. Removable in-tub seats shall be capable of secure placement. Permanent seats shall be 15 inches (380 mm) minimum in depth and shall extend from the back wall to or beyond the outer edge of the bathtub.

According to the IPC (International Plumbing Code) both bathtubs and showers shall conform to ANSI Z124.1.2 which only indicates none-mandatory bathtub apron height of 14 to 16 inches (Appendix II, Section II3.1.2). There is no shower curb requirement, thus curbless showers are legal.

That said, please confirm any requirements with your AHJ (Authority Having Jurisdiction), i.e. Local Building Department.