What is the definition of bribery (rishwa) in Islam, and is paying a sum of money to someone who arranges hotel bookings in exchange for giving the passengers to the driver considered bribery? If so, does this amount make the entire income unlawful (haram)?

If the booking organizer works at the hotel and takes money from drivers with the hotel's knowledge, without showing favoritism to anyone, then there is no harm in that. The prohibition lies in two matters:

First: Gifts to employees, which are what is given to an employee without the knowledge of their employer, potentially leading to favoritism towards the giver or overlooking their mistakes. It has been authentically reported from the Prophet, peace and blessings be upon him, that "gifts to employees are ghulul (ill-gotten gain/embezzlement)."

Second: Bribery, which is money given to invalidate a right or validate a falsehood, or to take what is not rightfully due, or to evade an obligatory right.