The other requirement of these redemption stories is that, for whatever reason, these characters do not die for their sins, and instead must live with them, and go on to the next stage of their redemption. Darla is an odd exception--in her sort-of redemption storyline, she twice (in "The Trial", and again in "Lullaby") reaches the stage where she is prepared to die. The first time, she reverts to evil against her will. The second time, she actually succeeds in killing herself.
(And then there's Connor, but since he doesn't complete a redemption story in the course of the series, I'll leave him out of it.)
Faith and Spike both emphasize how their badness while begging for death from Angel and Buffy respectively. Faith repeatedly tells Angel that she’s “bad” while begging him to kill her. Spike tries to convince Buffy to stake him on the grounds that he is an inherently bad person, and any good he may have done is merely “window dressing”. He also describes himself as a "bad man".
Angel, Anya, Andrew, and Darla, when expressing acceptance of redemptive deaths, may have made passing references to when “when I was evil”, or similar comments, none of them went on about how “bad” they were. Angel, when attempting suicide, was more concerned about the fact that he was weak than the fact that he was bad (“Look, I'm weak. I've never been anything else. It's not the demon in me that needs killing, Buffy. It's the man.”). Anya was more concerned that she had no idea who she was (“Xander, you can't help me. I'm not even sure there's a me to help.”). Andrew merely, finally, admits what he did, and makes the connection to Jonathan's feelings while dying and his own ("And I killed Jonathan. And now you're gonna kill me. And I'm scared, and I'm going to die. And this—this is what Jonathan felt.”). Darla for her first near-death mentions nothing about being "bad", and simply concludes that her second chance is to die, "the way I was supposed to die in the first place". Her final death in Lullaby is more hopeful: She dies telling Angel that their infant son is the only good thing they ever did together. She expresses horror about their past actions, but does not seem to consider herself inherently evil.
I assume Spike and Faith's insistence on their own inherent badness has something to do with their roles as Buffy's shadow-selves. They are the evil versions of Buffy, and she is the good version of them. In some of her darkest moments, Buffy projects her capacity for evil onto them. In their lowest moments, they project any capacity for good they might have onto her, and can't see the potential of themselves.