I have Barry's book but did not find the answers ...
You didn't look hard enough.
From page 162 of The DCS O Gauge Companion 2nd Edition:
Checking the Battery
The most effective battery check for a battery that's installed in a PS2 engine is to first start the engine up, making sounds, in either DCS or conventional mode. Then, immediately turn off power to the tracks. If engine sounds do not continue for 6-7 seconds after power is turned off, the battery needs to be charged.
Another test of the battery is to attempt to operate the engine conventionally. If the PS2 engine remains in neutral in conventional mode and sounds cut out when the direction button on the transformer is pressed, the battery needs to be charged.
The battery may be charged in three different ways. If the PS2 engine has a battery charging port, the most convenient way is to use the MTH Battery Charger (item # 50-1019). This charger will charge 8.4 volt or 2.4 volt batteries, either NiCad or NiMH. If the PS2 engine lacks a battery charging port, the second method is to place the engine on a track with voltage set at 10 to 12 volts to charge the battery. The least desirable method is to open the engine, remove the battery and place it in an external battery charger.
This and a whole lot more is all in MTH’s “The DCS O Gauge Companion 2nd Edition", available for purchase as an eBook or a printed book at MTH's web store!