So the term "Mini Cabs" are a result of confusing arising from weird local regulations in UK cities. The same as people think its normal that in NYC you can get a "Black car" which is different from a (Yellow) taxi.

But again: all these terms are just effects of the madness that is a local monopoly on one type of ride service (typically, but not always, the one that can use taxi stands and be hailed without pre booking on the street).

This has made people associate "taxi" or "cab" with just that one type of ride service, in their city. Which is understandable - but it makes no sense when discussing laws across a continent, or worldwide. "Taxi" has a very simple definition: it's a vehicle for hire with a driver. An Uber is a taxi service even if it's not (yet) allowed to perform the same services as a yellow cab in new york, or a london taxi