As far as I know, the 300 PLCs treat a power cycle and Stop->Run transition the same in the program. I'm not sure how it would look in the diagnostic buffer.
However, you could potentially look at the behavior of the devices external to the PLC. Something like a managed Ethernet switch will usually have a longer boot up time than the PLC will. If they are controlled by the same circuit and the PLC is plugged directly into the switch, you could do a test in OB100 to see if the port on the PLC is active or not.
In a run->stop->Run transition, the ethernet connection will never be lost.
In a power cycle, the port might still be waiting for the partner to come online.
You'd have to test the power on behavior of your system to see which components to monitor for your test.