The idea behind all this is that pilots making contact with a FINAL Director (ie the bloke who puts you on the ILS) make short calls so the frequency is kept as clear as possible. This achieves two objects 1) it enables the Director to make calls at precisely the moment he needs to to achieve tight spacing and 2) stops plane bumping into each other when they're only 2.5-3nm apart because the Director can't get a word in.
Unfortunately, the powers that be decided that the callsign of all controllers involved in the Approach Control function at busy airports have the "Director" callsign so pilots, understandably, can be confused. In a nutshell, when you transfer to s different ATC Unit, e.g. from London Control to Gatwick Director you should report cleared level, ATIS, aircraft type, etc. When you are then changed to the second Director you should use callsign only, as (hopefully) instructed by the first Director.