Okay, in regards to the difference between Gunfighting and Sharpshooting, both sides have it right in many ways. (WolfieGrey gets serious points for realizing that and also where the gunfighting bonus comes from after getting Miss Caraway to fire off her pistols at the target range; watch an artist at work and you are certain to improve your own technique)
Sharpshooting deals primarily with rifles where accuracy is needed because of the distances involved. Gunfighting deals primarily with close quarters combat with a pistol where being even a little slower than your opponent can mean your demise. However, there are numerous areas of overlap between the two skills and that is where things get a bit mushy.
Take the bear encounter for example. Sharpshooting excels here because although the bear is close, the killing target is small. If you don’t shoot the bear’s head or, preferably, the eye, then you’ve pretty much wasted your shot. A good gunfighter can put up a lot of lead and hit the bear’s center of mass pretty easily, but unless the critical target is pierced, that doesn’t mean much because the bear will kill you before it dies.
However, an excellent Gunfighter can also be devastatingly accurate. Which means not only does he or she skin out their pistol quick, but they can nail the beast’s eye as well. It is just harder than winging a drunk in a saloon to take him down quick. (That reminds me to add in an excellent sharpshooting result versus Hungry Snake in the Painted Caves. Hip shots via a rifle from a pro can be be executed quickly.)
Gunfighting also has a defensive component. Shoot first, yeah, but if you can’t shoot first then try not to get shot so you can follow up afterward. Sharpshooting doesn’t really cover that (Except in sniper fights), and is almost entirely offensive based.
As noted, Jesse James was an exceptional short range shot. He blended speed (Because if you don’t shoot first, you might not get a shot off at all), insight (Who to shoot first when confronted by a posse matters. Choose wrong and you might not get to pull the trigger again) and accuracy (Because it doesn’t matter if you shoot at the right guy first if you miss). I have no idea how proficient a shot he was with a rifle, but I would guess he was at least fairly good because, again, there is some overlap between the two skill sets.