The famous London taxis are actually called 'hackney carriages' but in London we just call them 'black cabs/taxis'. London taxi drivers have to pass a special exam called the Knowledge to get their license so if you take a black cab, you can be sure the driver will know the way.
Most of the English speaking countries use both words while the non-English speaking ones use taxi almost exclusively, or a regional derivative. Cab and taxi are two words we use to refer to a type of vehicle for hire with a driver.
The British English word for 'taxi' is also 'taxi'. The word 'taxi' is used to refer to a vehicle that can be hired to take passengers to a destination in many different countries around the world, including the United Kingdom. In some parts of the UK, the term 'cab' is also used to refer to a taxi.
Hack is short for hackney (from the Middle English hakeney), which at one time described a horse of average size, used for regular riding—as distinct from stronger horses used for hauling or in war. Such horses were often let out for hire to pull a coach or cab.
Taxicabs and other vehicles-for-hire in Canada are regulated by local municipalities and provinces, and are owned & operated by private companies and individuals. Unlicensed cabs in some cities are referred to as bandit taxis/cabs. Most cabs are large sedans with various colours and of domestic make.
What is correct in New York City, taxi or cab? Both are correct as the terms originated from the original word “taxicab”. Most people do call them taxis here.