Voltage being zero in a short circuit
This can only happen if you have an ideal battery. In the real world, nothing is ideal. You have an internal resistance of a battery, usually in the milliohm region, and you also have the wire shorting the battery out. That also has a resistance. What you have now done is created a circuit with resistance, and therefore current. You end up with something like this:







simulate this circuit &ndash; Schematic created using CircuitLab

This means that you will always have a voltage on the battery. The resistance will be very small, hence why there will be a lot of current flowing, which will mean a lot of power dissipated as heat, which is why it can be dangerous to perform such experiments!

As you can see, there is no such thing as a true short circuit, as there will always be resistance somewhere, no matter how small!

As for wrapping your head around voltage becoming non-zero when resistance is added... Well, you don't have to! For the zero condition to exist, the resistance must be infinite, if there was no resistance, then there would be no circuit!