A motor vehicle which carries passengers for hire or reward and is engaged under a contract, whether express or implied for the use of such vehicle as a whole for the carriage of passengers mentioned therein and entered into by a person holding a permit in relation to such vehicle or any person authorized by him in this behalf on a fixed or agreed rate or sum on (i) a time basis, or (ii) from one point to another, and in either case, without stopping to pick up or set down passengers not included in the contract anywhere during the journey and includes a maxi cab and a motor cab.