@tanguatlay,
Etymology
Harry Nathaniel Allen of The New York Taxicab Co., who imported the first 600 gas-powered New York taxicabs from France, coined the word taxicab as a contraction of taximeter cab. In time, the shortened term, taxi, came into common usage. Cab is an abbreviation of cabriolet, a type of horse-drawn carriage. In Britain, the word taxicab is rarely used.[citation needed] In the U.K., taximeter cab was shortened to taxi and cab, and these were and are used separately to distinguish between type of service and/or type of vehicle.[citation needed] In this article, taxicab appears throughout.
In turn, taximeter is an adaptation of the French word taximètre, which is a derivation of the German word taxameter, coined from Medieval Latin taxa, which means tax/charge, together with meter from the Greek metron meaning measure.[1]
So, the taxi is the method of payment, the cab is the form of transport.
Where I live you mostly have to call for a cab, unless you're at the airport, or maybe downtown.
I think "I'm going to call a cab" is more common than "I'm going to call a taxi", if only for the aliteration effect.
However, if I was hailing a cab, I would never call out "Cab!", I'd call out "Taxi!"
Calling out "Cab!" just doesn't seem it would get the desired result. Maybe because someone might think you are standing in the street yelling out "Cabbage!" or "Caberet!"
Calling out "Taxi" doesn't sound like anything else (except taxidermist), and is announcing you want to pay for a ride.