The difference for me between all those options is the motivations and what that says about the character. If he was traveling the world from a young age and learned those skills independently of being Batman it's not only telling of who Bruce Wayne as a person is, but also of his determination and natural talent. If he decided to become Batman afterwards for whatever reason and used those talents, that is childish, but we still get to see him be the mature level headed Batman, so we know that no matter how absurd the decision to become Batman was, he was not only capable but justified. It's still a stupid decision, but one that I can see coming from a rich guy with confidence that he could actually succeed on some level.
If he was a kid who honestly thought he could travel the world learning how to be a crime fighter, then he's just and idiot who tripped into success before someone could stop him. When a billionaire adult does it, he's an immature idiot potentially throwing his life away, but he would (or at least should) be aware of that. Said kid wouldn't have that self-awareness. For that child to actually follow through on that all the way to adulthood and become Batman make Adult Bruce Wayne seem less immature, and more delusional. You can hand wave it away with the fact that he's mentally damaged and the trauma of his parents death, but he's not really shown to be a mentally damaged person struggling with his parents death, or at least not struggling with it in a way that explains Batman.
That's ultimately my problem with his origin. I'm not a hardcore comic fan, but Bruce Wayne is a really smart, really talented, and outwardly well adjusted person (aside from the superhero thing which I can easily buy into because he lives in a world where that's just a thing that exist), who happens to have traumatic experience involving his parents death. Does that experience inform his character and explain his behavior as Bruce Wayne/Batman? Yes. Does that experience justify him becoming Batman? No, because he's rarely ever shown to be the psycho viewers paint him as to explain why that would make any sense at all. Sure it would get a pass back when origins just needed to give a bluntly relevant how/why, but now it (to me) it just comes off as bad writing getting a looked over.