Under EASA there will be no FTO, TRTO or Registered Facility. Every organisation that gives any training for any EASA aeroplane, helicopter, sailplane or balloon pilot licence, rating or certificate will be an Approved Training Organisation (ATO).
Don't confuse yourself by trying to compare ICAO and EASA standards - one is present, the other is future and, in any case, has not yet been finalised. As for LASORS - they haven't yet caught up with JAR-FCL, let alone EASA.
The current licensing standards, in the UK at least, are contained in JAR-FCL, as implemented or amended by the Air Navigation Order. Both of these documents will be superseded by EASA Part FCL/OR/AR if and when that legislation is finally passed.