I don't think DmC Dante wanted to kill him either maybe his devil side got the best of him
Dante didn't want to kill his brother in DmC. He tries to talk him out of it multiple times. Before the battle starts he's trying to convince Vergil he is speaking nonsense and says stuff like "don't make me do this." Dante always held the ideal to fight against oppression. It's why his sword is named Rebellion even.
Even the part when Dante gets the upper hand he goes "It's not too late brother." and Vergil just responds with a "Yes it is." and knocks him like 20 feet. That's showing that Vergil is
looking to KILL DANTE. He knows to stop his brother he may have to do something he doesn't want to do. At the end when he stabs him he's got no choice but to kill Vergil but Kat kind of snaps him out of that rage. He picks up his brother and Dante seeing his brother go off gets him depressed.
He doesn't think there is hope for Vergil. He's watching his brother leave knowing damn well in the future Vergil will return and cause problems down the line. He feels as though it's inevitable that he will have to probably kill his brother and he doesn't like that idea.
The original Dante's motivation was seriously just revenge. He even just freaking flat out says it in DMC 1. Devil May Cry was him trying to find the demons that killed his family.
Dante in DmC however isn't looking for revenge because he's just got nothing. He's not living really he's just surviving. When Vergil and Kat find him they give Dante context on his life. He understands more about what's really been going on. Why his life has been terrible and also that the entire world has succumb under the rule of demons. Sure Dante wants to get back at Mundus for personally screwing up everything he had good previously in his life but also simply because Dante wants to save the world.
He gives a sh!t about mankind and isn't just doing it because he wanted something to do on his weekend. Hell. he's fighting to save a world that actually sees him as the bad guy. He's public enemy number one and people are terrified of him. The world doesn't give him a reason to be a hero for them. He does though because it's what's right. Dante killing Vergil would be a choice that would have to be done for the greater good. Even though it pains him to lose him *le gasp* similar to the relationship they shared in the previous game. So I guess it's not so different after all.
In fact Dante's arc in DmC mirror's Sparda's Legacy and I can see making DmC a more Dante centric story arc would also mean maybe Dante would be the Legendary Hero instead of always living in his dad's shadow like he does in the previous series. Those games can't go 5 minutes without reminding you he is the son of Sparda.
My favorite Dante is from the very first game but at least DmC Dante is a more dynamic character that has more of an actual character arc here than any other version of him. Dante was originally about as complex as a coloring book and people really rose-tinted these characters. My personal favorite Dante is from DMC 1 but I'd have to say the more well rounded character is DmC's Dante.