These were called 'taxi-cabs'. Nowadays either word is used. In UK the word 'taxi' is used for the diesel-engine ones with a high roof to the passenger compartment (also known as 'black cabs/black taxis'), and the term 'minicab' is used for standard passenger saloon cars that just happen to be available for hire.
Although there are some phonetic differences, the growth of international tourism made the use of the word “taxi” more and more universal so that, nowadays, almost everyone can understand its meaning.
Traditional taxis are typically regulated by local governments, have fixed fare rates, and can be hailed on the street or booked in advance. Uber, on the other hand, operates under a different business model. It's a technology platform that connects independent drivers with passengers via a smartphone app.