How to differentiate between load and short circuit
For this application I would recommend a comparator (since all you care about is whether or not your millivolt threshold is exceeded and not the information in the signal itself).



Use the R1/R2 divider to set your reference voltage to a few millivolts.  You will want to use high-grade components depending on the level of accuracy you are after.  Alternatively you can change the divider out for a potentiometer which would allow you to set arbitrary thresholds.

In the inverting configuration, once the Vin is below Vref the output will go high indicating a "short".