And that is the point that we disagree.
ART 100 definitions (2008 NEC):
Branch Circuit. The circuit conductors between the final overcurrent device protecting the circuit and the outlet(s).
Feeder. All circuit conductors between the service equipment, the source of a separately derived system, or other power supply source and the final branch-circuit overcurrent device.
Outlet. A point on the wiring system at which current is taken to supply utilization equipment.
Not saying you are necessarily wrong but where is the outlet(s) located for a 50 amp single supply hot tub? If there is fuses in the control panel for the tub is outlet(s) somewhere within the unit or is the outlet(s) where field wiring ends?
From the way I read it the branch circuit is between wherever the outlet(s) is located and the last overcurrent device ahead of it.