I think 'hire' and 'rent' are both fine (british english)
Recently we rented a car and son, who loves cars, insisted on discussing it at length. To differentiate bet it and another we called it the 'hire car' (you can't have a 'rent car' though
) and son would look up at the roof in awe. Of course he was understanding 'higher car'. He still can't figure why our own car is not a hire car' as it is actually 'higher'.
