So does the green wire hook up to the battery as well?
The intent is for you to connect the green wire to the truck’s running light circuit somehow. That way the camper running lights turn on when the truck lights come on, and they are powered by the truck’s electrical system. You can use the running lights pin or wire for the trailer connection, but for some late mode trucks that will cause the truck to behave as though a trailer is connected. On my 2021 GMC I connected to the wiring for the fender marker lights to avoid this.