"It takes a cracked soul to cast." and "Murder tears the soul." just says that if you've gotten to the point where you could cast it once, that particular pre-requisite is already accomplished, so the work to crack your soul is already put in. It doesn't say anything about removing the requirement of wanting someone dead.

Though, so long as we're looking at evidence, if we take Quirrell at his word, then his ability to cast the spell despite not wanting his opponent dead is pretty strong evidence that the requirement is in fact removed. In fact, we already know that some "requirements" to cast spells are not set in stone: from that same scene, Harry cast the true patronus without the carefully practiced stance and wand twitches, instead merely "one desperate wish that an innocent man should not die -"—but the constant requirement in this case seems to be the thought that accompanies the casting of the spell, which is why I'm hesitant to believe the wish of death is removed from AK's casting requirement.