With the charger connected to the battery, it's actual output voltage and the voltage at the battery are the same. It is true that the output voltage needs to be higher than the resting voltage of the battery to charge it, but during charging that voltage is seen at the battery terminals. As a corollary, under heavy load with both supplying part of the load, the voltage of the battery, and hence the output voltage of the charger, will be below the resting voltage of the battery.
The battery charger's set voltage (the "charger voltage") is just what you're asking it to try to output. It doesn't necessarily actually produce that voltage under all conditions; indeed, in the cases you're talking about, it cannot.
If analogies help, it's somewhat like a cruise control in a vehicle towing a trailer. The cruise control setting is the voltage setting of the charger, and the actual vehicle speed the charger's output voltage, and the trailer speed the battery voltage. Assuming everything is set up properly, the trailer speed and the vehicle speed are the same, and can't differ without someone having a really bad day. If you go up a steep enough hill, the cruise control set speed doesn't change, but the vehicle and trailer speed will drop because the engine simply can't supply enough power to maintain that speed, despite the cruise control asking for more. In terms of power and gas consumption and distance traveled and so forth, the actual speed is what is important, not the speed the cruise control would like to go.
