The legal requirement for airframe and engine log books is take-off to landing time.
The legal requirement for your personal flying log book is the time during which you act as a member of the flight crew. As stated above it's from when you first start to move to when the a/c comes to a final stop. It's common to make it t/o to landing plus 10 mins to allow for taxying.
How the costs are calculated is entirely a matter between you and the person you hire it from. Tacho and Hobbs both have the advantage from the owner's point of view that he can check what you should be charged rather than relying on what you tell him.