'Hire' means there is specific compensation for carrying passengers or freight, but it is not necessary that the carrier makes a profit.
'Reward' means any 'valuable consideration', usually in situations where one person performs a useful service for some other person, and the person performing the service is given recompense (reward) for that service. It includes cases where there is no obligation to pay. Reward is not just an alternative word for hire.
While ‘hire or reward’ is not defined in law, the term has been interpreted through case law (what courts have decided when they've been asked to determine if someone has broken the law or not). The case law interpretation is very broad.
Amendments made in 2005 to the Land Transport Act 1998 incorporated case-law interpretations to the revised definition of passenger service, which now includes:
when passengers pay a specific charge that is expected as a condition of carriage, including part payments to cover fuel and donations
when the person or organisation providing the service receives specific funding to do so (eg a Ministry of Health grant to take older people on outings)
when passengers are carried as an integral part of, or reasonably necessary to provide, another service or activity (other than a transport service) for which payment is made.