OK, so then can he run #10, 3-conductor or 3 individually insulated conductors to a subpanel? I believe the cost to run 3 individually insulated conductors is much less, and physically more manageable, than underground-rated 3-c with ground. This third insulated conductor is termed a "grounding conductor". At the pool area, ground the subpanel with it's own rod and grounding electrode conductor, and tie all the pool metal parts to the same. The IAEI says in this case "the electrical system at the additional building or structure is treated like a service for grounding purposes" even though "the building or structure is supplied from a service in another building or structure by a feeder or circuit.
Also, isn't the pump motor "equipment grounding conductor" minimum #12? Lastly, although 2-c w/gnd is not acceptable, doesn't the cable still have to be suitable for underground, weather it's in conduit or direct buried. ....looking for your thoughts!
... Duane