Okay I understand what you're asking and I'm going to try to give an answer,...and this answer only applies to why it didn't just obliterate him in the mansion in the medical room or in the forest during Magneto's little pep rally.
If you have a gun in your pocket and you go out this weekend and you meet someone at the local bar or nightclub that you're interested in, you approach that person and that person is not responsive to your advances...are you gonna feel compelled to pull your gun out and kill that person?
Just because you own a gun it doesn't mean you feel the need to use it, you use it to protect yourself or to protect someone else,....unless you're just using it for personal gain.... or just flat out evil. The Phoenix is not out for personal gain nor was the Phoenix ever described as an evil being, it's not out to kill and destroy and take over the world, it has nothing to prove.... it just wants to survive.
Xavier said that the Phoenix is "a purely instinctual creature, all desire, joy and rage,..he (nor did anyone else) ever describe the Phoenix...as evil. Wolverine is that element that appealed to the Phoenix's desire which is also that side that appealed to Jean, the Phoenix knows that Jean loves Scott, but the Phoenix also knows that there's something about Wolverine that appeals to Jeans desire, and it's Jean's desire for Logan that the Phoenix tried to exploit and entice on that lab table, it tried to appeal to the side of Wolverine that is a purely instinctual creature, all desire, joy, and rage, that side of Wolverine that is the animal, that part of him that would be responsive to it's own similar instinct and desire. You guys are asking why didn't it kill Wolverine and my answer is why would it kill Wolverine? It saw no need to kill him, although Jean may desire Wolverine, Jean doesn't love Wolverine and the Phoenix knows this, Wolverine was not perceived to be a threat to it's existence,...Scott and Xavier were.