If the quest is active in the game, meaning the maker of the game hasn't completed it. No matter who deals the killing blow, aslong as you're in the area where Diablo is, quest is solved.
Sorry, but that isn't the way it works for Diablo and gaining access to the next level. From my (admittedly limited) experience, even if you are in the area, and the quest is active, if someone who has previously killed Diabo kills him this time, you might not get credit for the quest. On a couple of occasions, trying to build a fast "sir" to enable HC on an new account, the quest was active and the "sir-to-be" was in the area, but we had to do it over because he did not get credit. On other occasions, "sir-to-be" did get credit. My guess (and that's all it is) is that, in addition to being in the area, "sir-to-be" must do some damage to Diablo. I know that he does not need to stike the killing blow.
On at least one occasion, however, the "sir-to-be" was dead at the time Diablo was killed, yet got the credit for the quest. Others have reported that if the character killing Diablo is doing it for the first time, everyone partied with that person gets credit for the kill, even if in the PF at the time.
So, the situation is not that simple. I don't know what all the requirements are, but I do know that they are more than you list.
