However you are charged should be built into the price. Some will charge hobbs, but the hourly rate may be less than a school which charges brakes off to brakes on, or even Take off to Land plus 10 or 12 mins. This allows for the time you have the engine running on the ground which you cannot log.
This does make it difficult to make a proper comparison of hiring charges, but that's life.
Unless you actually took to the air, there is nothing to log. If you did an extended ground run, this may be worth entering in the engine logbook, but certainly nothing for airframe, prop or pilot.