I can quite easily measure the resistance of under 1 inch of meter lead… it just takes the proper equipment.
Most cheap meters have at least 0.1 ohm shunts on the high amp ranges. The problem with measuring tail cap current with them is the voltage drop across that shunt (ohms * volts) is a very significant percentage of the battery voltage. Couple that voltage drop with the extreme variation of LED current with changes in voltage and you can easily get a huge (like 50%) error in LED current without the meter burden resistance. The total shunt resistance in my LED analyzer (Any interest in a LED/Battery analyzer device?) is around 2 milliohms (50 times lower) than a typical meter.