A shit load of tricks in all forms of action sports have multiple names. Mostly people just not knowing enough and starting a trend is how confusion comes about. Freaking skiers saying their going switch when it is fakie is the most wide spread example. Cab comes from skateboarding from Steve Caballero. Half cab is a fakie 180 and full cab is a fakie 360. Since on sowboarding you can't change foot position fakie and switch are the same thing (mostly). Caballerial - Wikipedia, the free encyclopedia
In skateboarding, a cab trick means riding backwards, and doing a frontside spin while popping off the TAIL, whereas a true "switch" trick means popping off the nose, as if you were truly riding the opposite way.
But people have lost/not understood the distinction, so nowadays cab just usually means "switch frontside"
True "cab" spin butters and spins are fun on flatground tho.
A cab spin in snowboarding refers to a switch frontside spin. Switch backside spins are called just that. The thing is though using the the term cab doesn't really make sense in snowboading. The term came from skating and was a fakie frontside spin. In skating a fakie frontside spin is arguably the hardest spin because your riding fakie and spinning frontside which is generally to your weaker side. In snowboarding you can't really ride fakie because your feet are strapped in and therefore your feet can't shift. The way i think snowboarders picked up the term was because a switch frontside spin is arguably the hardest way to spin on a board so the just placed the term cab on it.
Now that i think about it your right as a fakie ollie would be same thing as switch nollie. That being said i hear most people still refer to it as a swtich frontside spin even though for it technically to be called a cab it would need to be a nollie as Extremo said.