I think it goes by either name, and I tend to agree, but I just thought it lead to no consequences in Wagon's game. No one's name was revealed, blessed wouldn't stop it, brutal or vengeance wouldn't trigger (I don't think), so it just lead to this feeling that if you were targeted by an assassin, there wasn't anything you could do, and if you were an assassin, and decent at your job, you could murder with impunity with very little risk.
Though, now that you mention that you are using the alternative name for it, I see that it does reveal the hunter's name should they choose to use it, which I think leads to a fairer balance in the game and a good source of risk before pulling the trigger.