Subpar Spatula
Banned
It's a matter of intent. T'Challa didn't hunt Bucky because he believed it was what the law wanted. The law wanted Bucky to be transferred into custody and then extradited to Wakkanda. T'Challa wanted him dead.
Google defines "Vigilante" as
Whereas Dictionary.com has both
Bernthall's Punisher fits both definitions at different times. Punisher's motivations determine which definition of vigilante he fits. When Punisher kills people like the guy peddling pedophilia, he's not doing it because he has an axe to grind. He's doing it because he thinks the law has failed and he's dishing out what is deserved, When he goes after the Irish though, he's avenging the death of his family. He's appeasing his own rage. These are two similar but distinct definitions of "vigilante." T'Challa happens to just be the 2nd. The Avengers though, they don't go out on personal missions, or at least they're not supposed to. Tony took a step into the other Punisher territory where he wanted to kill someone out of rage. His intentions have always been good up to now, even if his results often blow up in his face. He went straight for murder this time, as did T'Challa.
I mean, it's all supposed to be a theme of centered on actions taken out of anger. Zemo did what he did because he was pissed at the loss of his family. He hated the Avengers for what happened and blames them. T'Challa was enraged enough to kill on sight for the whole movie. Tony loses his shit and stops being a hero. All those actions were driven by rage. T'Challa just happened to be the one who broke the cycle that demanded murder in response to murder.
Your understanding of the word is wrong. Vigilantism is taking the law into your own hands. This is made very clear when Black Panther tells Zemo that the living still have need for him because he's bringing him to justice now and instead of killing. You don't even need that bit at the end to make him a vigilante because instead of going to the authorities and letting them handle it, he got in costume, and chased down the man who "killed" his father. This is vigilantism. Intent has nothing to do with it except to justify the vigilante's actions.