A close look at the high power LED shows 16 dots that are parallel little LEDs on the substrate. At the rated 350mA then each of the little LEDs draw a current of 350mA/16= 21.9mA.

 

I have to ask. You don’t have a resistor in series with the LED, do you?

I suspect that in your experimentation, you damaged it.

 

Seems to be some confusion here.

First, the X-Lamp LEDs generally use a single large die, there are not 16 small chips in an XP-E2, the dots are the surface treatment, part of how the device is manufactured.

Second, the green unit has a Vf of typically 2.7V at 350mA (not 2.2V), but that is a typical figure, the Vf and other parameters vary depending on the binning codes. I would expect to see what you saw at around 3V as there will be a small voltage drop in the leads between your power supply and the LED, so the LED would have been seeing less than 3V and so raising the voltage will of course raise the current as the voltage travels along the I/V curve.

So, you are worrying about nothing, what you are seeing is perfectly reasonable for this device. You need to measure LED Vf at the LED, not rely on the power supply's meter.