Both are correct, regardless of whether you’re speaking British or American English. If that’s all you wanted to know, you can now go about enjoying your Sunday. But if you want to know how goats, Greek, a sign of the Zodiac, and stuffed animals fit into the equation, then please, read on.
The word cab is older than taxi. The latter is short for taximeter cab. This term came into use in the early 20th century, taximeter referring to the newly-developed meter used to automatically record the distance and cost of a cab ride. The taxi- part comes from the medieval Latin word taxa, meaning charge, which is also where we get the word tax from.
As I was thinking about the word taxi, I got to thinking about the word taxidermy, the practice of stuffing dead animals. I wondered if there might be a link between the two similar words. And the answer is: maybe, but probably not.
Taxidermy is decidedly Greek in origin, being derived from the word taxis (order) and derma (skin). It therefore means ordering of skin, which is obviously something a taxidermist does. There might be some tenuous link between the Greek taxis and the Latin taxare, but alas, they’re probably independent.
That’s the history of taxi, but what about cab? It’s short for cabriolet, an 18th-century French word, itself derived from the Italian word capriola. This word referred to the exuberant leap of a small deer or goat (capra in Italian, like Capricorn), because the suspension on cabs was quite springy, and they’d “leap” on bumpy streets. Capriola means somersault in modern Italian, and is also the origin of the English word to caper.
Feel free to share all this with the driver on your next taxi ride.