All of what you're all saying are true...I think for an aircraft to be designated a bomber would also require size and a much greater payload. Raptor probably explained it best - it handles like a fighter and if given more powerful engines, could probalby evade enemy aircraft, but then again, would take away from its stealth capability.
Look at the F-15E Strike Eagle - it is a fighter and ground attack, so why not designate it F/A-15E - I would think obviously for simpicity's sake. But it performs the same role as an F/A-18 Hornet, although I would think a lot better.
The Air Force is also looking to retire the F-117 early, mostly to receive more funding for the F-22 and F-35. I think they should keep it in service because it can still evade radar, but with the technology they have now compared to 20 years ago, I think the Air Force can re-equip the F-117 with more powerful engines capable of supercruise. It doesn't necessarily have to go Mach 2, although that would be an added benefit, but as long as it can achieve Mach 1.6 in supercruise, it should be sufficient to evade enemy aircraft. What the heck, why not add an internal cannon for self defense?