What voltage do we use for battery cut off voltage?
The assumption is that you are measuring the battery voltage while it is connected to the load. It isn't practical to expect someone to periodically remove the battery from the load to measure its voltage.

As for your Sony battery, you haven't really told us what kind of battery it is or how much current the load is drawing from it. We can't give a good answer without that information. You may be overloading the battery...trying to draw too much current from it. Or, the battery may be faulty.