It is probably better to think of the BAT switch as a Battery Isolation switch which controls a contactor of some sort. Your statement appears correct.
The output from the Stator goes to the rectifier pack which is internal, so a single DC output terminal is normally provided with the frame as ground. This would normally go directly to the main Busbar. The battery would normally be connected to the DC bus, via the zero reading ammeter in Cessna aircraft. The manual may well show only major components.