Actually Goku Black is more like Ginyu, Baby is outright possession.
It's more like a fusion, so he'd still be a half since I counted Vegito and Gogeta as halves.
They really took Vegeta saying Goku is better to mean that Vegeta just kind of... gave up training.
Which is totally out of character.
I don't think that's the case. He's still shown to have his training equipment, and he has that conversation with Trunks and Goten about how they've skimped on their training in these times of peace, implying that he, himself, hasn't.
I think they ran with the idea that Vegeta, while still motivated to train, had truly recognized that Goku had something he didn't. We have to remember too that in the context of GT, set fifteen years after the end of Z, Vegeta hasn't seen Goku in five years, nor has he had literally anyone at his level to fight against or threats to thwart. He's become a family man, apparently making his own money as he says Bra is spending all
his money, which means he's no longer a freeloader training all day every day. They took that moment where he sacrificed himself against Buu, and then the following moment of internal recognition of Goku's superiority, and made it a turning point for the character into his true maturity. People change as they get older, and unlike Super's Vegeta, GT's Vegeta had fifteen years of peace in which to think about these things and change his ways.
Where Super alters this is by introducing a new way to level the playing field for Vegeta-- the SSG transformation that's given just by having five saiyan friends-- and reverting him
back to how he was prior to those revelations at the end of the Buu saga. He and Goku are even again, they have divine ki now, and he gets to skip SS3 and challenge Goku anew, reigniting that spark. Since this all happens within, what, a year or two of Buu's defeat and before Bulla is even born, it's believable. But I'd argue that so is GT's characterization. People change. They realize things about themselves. And in both Super and GT, Vegeta turned from a manchild obsessed with being the prince of an extinct species to a father and husband responsible for his family-- they just went about it two different ways across two believable paths.
EDIT: A further note that I need to praise GT for: The implication that Vegeta can't reach greater heights by
training alone.
Aside from his powerup to SS1 offscreen, born of obsession with Goku achieving it, powerups happen either through training with others or story-explained reasons or needs produced by a more powerful threat. In the context of GT, Vegeta had
none of those for at minimum five years. If we were playing by Super's rules, he could just "train" offscreen and become a god or whatever. But by stricter DBZ rules, Vegeta had no catalyst to cause a change, no "need instead of a desire". Like the humans who reached their peak, Vegeta had reached his. And nothing was around to propel him past it after Buu was vanquished. We have to remember that Goku willingly said bye to his seemingly only equal to go train alone with the reincarnation of Buu. Ten years after Buu had died. It's possible that both Goku
and Vegeta were simply running idle, unable to get any stronger with each other until Baby came and forced both of them to. One through desperation and one through artificial powerups.