The AUTO part is easy to understand. In that mode, the lights are trying to light at the same time as the engine is cranking. HID lights need a very large voltage in order to establish the initial electrical arc. Obviously, the period when the engine is cranking is the very worst time to try to get a very high voltage. A very small delay between the time you press the START button and the moment the lights turn on, say a second or less, would probably help quite a lot in resolving the problem. The OEM system might just be adopting a similar solution.
As for the problem happening when the bulbs haven't been used in a while here is my theory. To understand why, you first need to understand how HID bulbs work. If you look closely at an HID bulb, you will notice a white/yellowish powder is in there. That is a salt that gets vaporized by the initial (very hot) electrical arc. The initial arc requires a LOT of juice to produce because the thin gas inside the bulb is not very conductive. The arc is also initially blue. Once the salt is vaporized into the bulb, the gaseous medium becomes a lot more conductive and better able to sustain the arc. Ionization of the salt vapor by the arc is also what produces the white light color we all love.
Now, after turning off the car, it takes a while for the vaporized salts to re-condense into a solid. During that time, the gas inside the bulb remains more conductive, so a lower voltage is not as much of a problem: the lights should start easily even in AUTO. Once the bulbs have cooled completely and salts are condensed, then you get the problem agin on occasion...
I think your theory is correct. I only turn on the light after the engine crank has die down now but sometime it still won't turn on, I will wait a bit longer and see if the light will turn on more consistently. Would be nice if TRS would make the relay with time delay, this way it would be sure to work if the light switch is in AUTO. Just wondering if anyone here with OEM HID, how long after you start your car does the light come on?