Taxi, taximeter, taxicab & taxidancer

The Online Etymological Dictionary (www.Etymoline.com) gives the following
origins of the word taxi and its derivates:

taxi 1907, shortening of taximeter
cab (introduced in London in March 1907), from taximeter "automatic meter
to record the distance and fare" (1898), from Fr. taximètre, from Ger. Taxameter
(1890), coined from Middle Latin 
taxa "tax, charge."

The verb is first
recorded 1911, from earlier noun use as slang for "aircraft."

Taxicab is also first attested 1907. Taxi
dancer "woman whose services may be hired at a dance hall" is recorded from 1930.

WorldWideWords.com
supports this explanation by stating:

“The
German name of Taxameter, at first adopted in Britain, was taken from  Taxe, a charge or levy. After the
device became common in Paris , the French created the term  taximètre for it, from  taxe, a tariff. 

For the complete explanation provided by
WorldWideWords, follow this link.

Cab, a shortening (1827) of cabriolet,
"a two-wheeled, one-horse carriage." This word, first found in
English in 1766, came from French cabriolet, of the same meaning, which in turn
was derived from cabriole, "caper," because the vehicle
moves along with a springing motion. [Italian  capriolare "jump
in the air," from Latin  capreolus, meaning  "wild goat, roebuck."] Cab, the shortened form, was applied to other vehicles as well,
including eventually public conveyances.

Taxi Marseille – English classes – cours d’anglais :