Brand new 9v shows as ~11v read directly, and 6v again on arduino.
No matter how inaccurate your meter is it will not read higher than it is supposed to for one reading and lower than it is supposed to for another. Your readings indicate that your Arduino is drawing excessive current which is lowering the voltage delivered by the battery.
To see if this is what is happening you can try the same set of measurements with the Arduino.
Don