Actually, the textbook would be wrong. Hansom Cabs began operating in London and Paris early in the 19th Century. The word Taxi is short for Taximeter Cabriolet, a vehicle used for transportation introduced in New York in...err...I believe the 1840's.
Essentially, this whole American/British English debate is odd. The British colonized America, then got royally slapped. Daniel Webster produced the first American English Dictionary. A few alterations of spelling, a whole lot of slang phrases and popular idioms and that is it. I make a point of telling my students that, if one removes the regional slang argot, youth-culture lingo and idioms, the basic difference between English as spoken in the US and that spoken in the UK is spelling and flat/long vowels.