The phraseology is linked to the title of the chart - which is currently RNAV. Request RNAV approach Runway XX, There could be three minima lines on the chart, LNAV, LNAV/VNAV or LPV but ATC don't need to know which one is being flown because the procedure is identical apart from the DA/H. The pilot needs to know which minima line he can use.
I think that for Baro/VNAV procedures in France there is an addition read back of the QNH - because if it was wrong it could be rather important.
There is a move to change RNAV approach chart titles to be RNP but this will take a while. All these approach types rely on the use of GPS which includes onboard performance monitoring alerting so the RNAV approaches are really RNP approaches according to the definitions in the ICAO PBN manual.