You got it right for most of the C class motorhomes. When the engine is running it charges both truck and house batterys. When hooked up to shore power the charger charges the house batterys. The dash switch will let you start the truck with the house batterys if it is down and house batterys up. Your house generator will charge only the house batterys only in most cases.
When pluged in some chargers will cook the batterys. Others will care for them and just go to standby when charged there are a number of brands so no way to know what you have with out looking at it and reading what you can.
If you did not get the manuals for the stuff in your rig they are on line. Post if you need them and I will post the address.