Once, after a victory, when an essedarius was applauded for setting free the slave who was his driver, Caligula was so annoyed at the gesture that, leaving the amphitheater in a huff, he tripped on the fringe of his toga and fell headlong down the steps, fuming that "The people that rule the world give more honour to a gladiator for a trifling act than to their deified emperors or to the one still present with them" (Suetonius, Life, XXXV.3).
Claudius, on the other hand, was less petulant. When the four sons of an essedarius pleaded for their father's discharge, he granted him the rudius (the wooden sword signifying this release), circulating a note among the applauding crowd that they should desire children, themselves, "since they saw that they brought favour and protection even to a gladiator" (Suetonius, Life, XXI.5).
"Chariot," itself, derives from carrus, which, in turn, comes from the Celtic word for a wagon or cart. From the Latin is the etymology for such words as "car," "carry," "cargo," "caricature" (from carricare "to load"), and "career" (referring to a pathway or course).
References: Suetonius: The Lives of the Caesars (1914) translated by J. C. Rolfe (Loeb Classical Library).