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So I have a character in a dystopian novel I’m working on by the name of Tyler. He’s inspired by Alex Høgh Andersen’s portrayal of Ivar the Boneless, and his death... is interesting. He’s, for the most part, an antagonist in the story, and I really felt bad about how mean I was to him, so I created a character to love him. Now, my first thought was to create a character that was a love interest, but with characters like him, love interests are often manipulated and exploited by people like him. They have their own reasons for being with the aforementioned person, and that could get messy you, so I decided that this person would be a child. Children live differently, in my opinion. It’s a rather unconditional love, a love that just feels differently, and he’s never had that. He’s never had someone love him simply because they do. His mother loves him, but only because she can use him. His brother’s love him, but only because he’s their brother. His father loves him, but he feels like he has to. But his niece just loves him. She doesn’t love him out of pity or obligation. There is no rhyme or reason why, she just does, and because of this, he is very protective of her, and unbeknownst to him, her fate is now aligned with his, because he will kill any who hurts her.

The person who hurts her... he makes good on his promise, but through that, he incurs the wrath of his father, and he is executed for it. I don’t think it’s a redemption arc because despite it all, he’s a horrible person, has always been a horrible person, and nothing about that ever changes. But he does a good deed(At least as good of a deed as that can be), and I feel like some people would appreciate that, especially his brother. But... that doesn’t feel like redemption to me, even though I’ve been told that it is. But maybe I’m wrong.

I would not call it a redemption arc, I could see this as simple revenge for somebody taking some piece of property he was enjoying.

In a redemption, the character realizes they have been wrong and becomes a better person. This sounds more like a character that promised consequences, delivered them, but made a mistake in doing so and suffered the consequences of that mistake.

Straight up gangster stuff out of the Godfather, very similar to the death of Sonny Corleone taking vengeance on behalf of his sister. (Sonny is the hotheaded brother of Michael Corleone, the MC).

But Sonny wasn't "redeemed". He doesn't harm women and children, but he is still a brutally sadistic killer and proud of it. He's never on the road to being a good guy, or moral, or anything more than a criminal. He just let his temper get away from him, and made a mistake that cost him his life.

It sounds to me like you have a complex character. Not a good guy, not a good person. Just a person with complex motivations and someone he'd die for.

It doesn't sound like a redemption arc, because he doesn't change his ways. All this is, is a man sticking to his core motivation, and the consequence thereof costs him his life.

A redemption arc, in most cases, is 'siding with the protagonist' or, at the very least, turning good--whether objectively or simply in the eyes of the viewer/reader.

This feels to me like someone real. Had a hard life, yeah sure. Look at Snape from the Harry Potter series. You can view him as a hero, but he wasn't. He was a douche, through and through, that did a few things right.

Whether it's redemption or not depends on what made him a bad guy to begin with. Was he the kind of antagonist to brutally slaughter anyone who slightly wronged him, and anyone who got in the way of that slaughter? In that case, avenging his niece would just be more of the same- hard to find redemption there.

Was he a cold, loveless loner who cared nothing for anyone else, using them only for his own advantage and discarding them when done? Then avenging his niece, something driven by his care for her and that he knows will severely disadvantage him, might be a form of redemption.

Redemption isn't one-size-fits-all; it only works if it's counter to the character's established villainous behavior patterns.

How does knowing the proper name of the arc you are narrating affect your writing? It seems to me that you aren't actually asking a writing related question but one related to the interpretation of an existing text that should rather be asked on Literature.SE.