Probing With Multimeter Breaks LED - Happened Twice
You need to break the circuit and put the meter in SERIES with the LED+resistor+battery.
It will measure the same current wherever you put it, since it's a series circuit.
You are putting it in PARALLEL (across the LED) which means the LED is shorted (almost) and the current comes from the battery + resistor so you get (ideally) 90mA = 0.09A not 0.009A with 100 ohms (maybe you used a different value of resistor like 1K).
When you put it across the resistor, the current flows through the meter and LED with little resistance so the LED is damaged (can also blow the fuse in your meter). Some meters can be damaged by this (but the Fluke is a good product and is well protected, still it can be a pain to have to replace the fuse).
A cheap meter used this way across the mains can be physically dangerous. Some meters have an alarm beeper that goes off when the probe is inserted into the current measurement jack and the switch is set to a voltage range (it is easy to forget and stick the probes across a power supply, so it's a good habit to always return the probes to the voltage jacks).