I can't begin to list all the ways that second story is better. First, Selina's such an individual in it, one who spent her life training for the job she has--something I always love in Batman stories where these people aren't granted super powers. It's like somebody actually thought about what kind of girl would grow up to be Catwoman. She makes age-appropriate mistakes, too. And she's barely inspired by Bruce. She's more defying him early on--like, if you think Bats are great, wait until you see a Cat.
Also I'm struck by how that second origin so clearly marks her as a parallel and equal to Bruce, the way both of them are driven and focused from a young age to their goal. Plus, I'm a bigger lover of thief characters and this gets that she's a thief rather than a supervillain who happens to wear a cat costume so we'll call her a catburglar. As you say in the commentary, it doesn't get less catburglar than beating people up and stomping in to steal things. If you happen to watch the show Leverage, Selina has a lot more in common with the cat burglar character Parker, but they're having her act like the muscle of the group, Elliot.
Finally, I'm a sucker for using the animal to characterize her. I love the way it takes things about cats and shows why she'd be attracted to them, especially the way it's not just the sexy personality traits but ones that would genuinely make people dislike her. In my head I think I always start with some sort of totem idea for the early crew--Bruce=bat, Selina=cat, Dick=Robin (even if that last one could have originally meant a different kind of robin).