Im not going to read all the previous posts, but
If you want to know for sure if the gen charges the battery
While on standby, then disconnect the battery from the gen.
Connect a small 12v light bulb to the GEN wires that go to the battery.
Note dont connect the bulb to the battery, connect it to the gen.
If the bulb lights, then the geni charges the battery,
You could use a meter with the bulb to verify the voltage.
Using the meter without the bulb could give you
A false reading because a modern dvm
Is a very very very small load. Thats why the bulb.

Mark