In the most techincal definition, a Streetfighter is more of a bodywork downgrade. The Streetfighter craze was born from hooligans who rashed/crashed their sportbike. Rather than spend money on new plastic, they took it off. Since most sportbikes have their lights in the plastic, they had to come up with something to slap on the forks to light up the roadway, so they either put on headlights from standard (aka, naked) bikes, or some kind of headlight in a fairing sort of deal. As this became a more popular way to fix a damaged bike, people started doing them up as customs.
In a way, the B4 (except the faired "Limited") is a factory Streetfighter, similarly, the naked B6 & 12, Triumph Speed Triple & Speed Four, Ducati Monster, etc. are - just one difference, when most factories make one, they detune the sportbike motor of their hooligan bikes.
To see a wide variety of Streetfighters, check out
http://www.streetfighters.nl/