Suppose it all depends on the Flying School but surely the same company wouldn't have two different 'time' charges for two different helicopters?
Actually, some do. All the ones I know of charge purely by Datcon time of both machines.
In the R22, as stated, the Datcon starts when you switch the engine on, and stops when you turn it off. In the R44 the Datcon only turns when the collective is raised...so you don't pay while you're warming up, shutting down, or in autorotation.
Adding a little to the R44 charges, like 0.1 hrs at start and finish, is fairly common.
Most people log their flying hours the same as the Datcon hours. Since it's rare to qualify in 45 hours, the possible slight difference in the R22 won't really bother anyone. For the R44, I usually use my watch and log the "real" flying hours.
Hence the R44 is slightly cheaper than it appears, but it'll never cost less than an R22!
However, although this is all very interesting, it's probably the least important question when you're learning to fly. The school's reputation, how well you get on with your instructor, how often you can fly, what books to buy etc etc....these are all far, far more important than whether you save....the price of your next hover-taxi to the fuel bay - and yes, in many places you'll pay for that too.