The fixed fare structure I linked runs on a meter - there is no interpretation possible. And the structure rewards drivers for taking a higher number of shorter

The fixed fare structure I linked runs on a meter - there is no interpretation possible. And the structure rewards drivers for taking a higher number of shorter journeys to avoid the situation you are describing. Paying a third party cash before a journey is arranged is not the typical use case for a black cab. In a traditional hotel situation, the concierge would either wander out into the street to hail a taxi for you that you pay for yourself (black cab), or act as a booking agent for a taxi (mini cab). It sounds like your family was being ripped off in the second scenario, and possibly tricked into thinking they were taking a black cab.