Yes, he is suggesting that (although not entirely accurately) and no, an hour invoiced is not necessarily an hour's flying.
Some aircraft work on a switch on the collective and others on engine time or rotors running time. These times are important for checking the maintenance scheduling. However, how a school charges is largely up to them! There are Datcon meters, Hobbs meters, all sorts which measure various running times.
However, the time you put in your logbook for training purposes is as defined in LASORS - download your copy now!
Cheers
Whirls