But Faith did not consciously put a stake in the heart of a person. Her decision to swing the stake was based entirely on the belief that the target coming at her was a vampire and would not have swung if she'd known he was humanThat's a totally different situation from consciously putting a stake into someone's heart. The lack of knowledge that she was staking a normal person doesn't take away from her intent to kill. That's why I labeled the act against Finch as "manslaughter" instead of "murder". It's like accidentally killing a person when you stab a trashbag that you thought was empty, but actually had a person hidden inside. It's still manslaughter by definition of the term, but it's done without the intent to kill a normal person. The laws charging someone for taking somebody else's life are there for a reason.
Even if I'm to pardon Faith for having made a mistake, there's still the consideration of how a murder changes your perspective. Faith might find an excuse to kill human murderers that are a harm to community. It's like what Angel said; she's had a taste of blood. Everything changes when you kill.
Vampire-slaying is a very morally grey area because there's no other way to compromise when it comes to putting down a vampire. No laws can be used against vampires; the same couldn't be said for Finch. That's why what Buffy does is the best moral compromise any one could have in her situation; Faith's actions were not.
Also, I would like to reiterate what Faded90 said:
Even if she isn't imprisoned, the least she needed was to be rehabilitated by Giles and Buffy. Same path, more effective solution that leaves less room for Faith to be abused by the system.
I reemphasize once more: putting Faith either in prison or under Giles' guidance towards a less murderous path isn't about punishing Faith; it's about helping her. Faith denied both solutions because she was lost and confused at that time, but it's clear that some form of preventive measure was needed, be it prison or Giles' guardianship.
Faith turning herself in was after she was no longer a threat to the community. She has accepted her crimes and was ready to begin redeeming herself for her mistake. It wasn't about punishment either at that point, but about redeeming herself for killing people and becoming accomplice to the murders of Sunnydale High students in "Graduation".
In spite of what the governments of America might think about the purpose of the American justice system, I personally believe in the rehabilitation of criminals, so "punishment" is never associated with imprisonment of criminals for me. It's either about the rehabilitation of remorseful criminals or as a preventive measure against non-remorseful (and mentally ill) criminals. For Faith after she turned herself in in "Angel", it's more about the former (rehabilitation) than it is the latter (preventing her from harming people). Granted, the real world doesn't work that way and the government often abuses the system and mistreats inmates (especially female inmates), but it's the best we can do with the system for now.
In fact I don't believe Faith either developed a taste for blood or thought she was right. I believe she acted the way she did in search for someone to tell her everything was going to be okay and she'd done nothing wrong