Those terms are not used in the US, you'll have better luck in another forum. Or you could just try Google "TRE", first thing that came up was instruction for TREs.
In the US, any CFI with a type rating can give instruction towards that type and airlines can designate ATPs as instructors in their own programs without the CFI. If an airline can show the FAA can not supply an inspector to meet the needs of the airline training center, they can request that someone on the training center staff be designated as a Aircrew Designated Examiner. A 142 training center can do the same thing but they are called Training Center Evaluators in the 142 world.