"No person has the right to initiate force, threat of force, or fraud against another person or their property."
This is what the NAP says.
"No person has the right to murder someone else unless that person is trying to murder someone."
This is not what the NAP says(even in the specific case).
You are assuming that both the aggressor and the aggressee is accepting the NAP, which i tell you: they are not.
It doesn't really matter if someone is not accepting the NAP if they live in a NAP society. Just as it doesn't matter if the person doesn't accept laws against murder in a statist society. They will attempt to murder regardless of the laws/rules, so we can ignore this.
For the rest, let's play out the scenario, assuming we live in a NAP society. Remember, NAP is just an agreement, not a set of laws imposed by any single body. So...
You see someone who doesn't care about what is right, threatening to murder someone else (initiating aggression). The victim is powerless to do anything. So, you step in and stop the aggressor, either by subduing them, or by killing them first. It was the initial wannabe murderer that initiated the aggression, not you. Since there are no specified laws, what do you suppose other people around you will do? Chances are, they will agree with your action, and applaud you for coming to the rescue of the victim. Again, there are no laws or rules regarding this, just general agreement, so what will result is literally what you believe others will likely do. Since people are generally good and fair, the likely outcome will be that as well.
Now, what do you think will be the outcome in that exact same scenario in a statist society? There are specific laws that state that only the police, or other people with power, are allowed to defend and kill. So, after you step in to protect the victim, even if everyone around you agreed that what you did was right and necessary, you still broke the law, and the first thing that will happen is that you will be arrested. Afterwards, you will be investigated, and if lucky, released with a warning. If unlucky, you'all be charged with manslaughter or murder.
Tell me I'm wrong.
In statist society, murder is allowed, too.
yes, under certain circumstances.
No, under all circumstances. Murderers can, will, and have murdered, regardless of what laws or NAPs or general agreements there are. Laws don't protect from psychopaths. That's why we have weapons.