A Line Trainer is like a line check airman in the US in that they do training in the aircraft, such as OE and line checks. A TRI
is a type rating instructor. They do required training in the simulator towards a type rating, but do not have examining authority. A TRE is a type rating examiner, who does have the examining authority required to issue type ratings.
This is my understanding, although I am an FAA rated pilot, and our terminology is a bit different. I did go through CAA conversion training for my UK CAA ATPL in the 757, and I believe I am correct. However, any CAA/JAA pilots out there may want to correct me or give additional information that I may have left out.