It's context in the #0 issue of Teen Titans, and Tim definitely held off on the Wayne moniker and adoption for at least one solid year and then some in Pre-Flashpoint continuity. And even then, he really only got obsessed with the name when Bruce died again, which makes sense, as that's his fourth parental figure in that history to bite the dust; you'd want to hold onto some stability under that trauma, and the name acknowledges his brotherhood to Dick and the rest of the family.

And the #0 issue seemed to be more the car wreck of an over-worked Lobdell. He'd just come off the bloated Culling arc and was prepping for his first few issues on Superman while still writing Superboy and Red Hood and the Outlaws. I expect him to improve the story no matter what in Secret Origins.

But the context of the issue wasn't good. Bruce doesn't want a Robin...but he's still stalking Tim. Tim's an Olympic level athlete, thus trying to steal some of Dick's thunder... But the event he's shown competing in literally doesn't exist. He's also supposed to be an unbeatable hacker...but Batman basically trolls him in a manner that seems somewhat stupid. He then displays enough skills to steal millions from Penguin...but is too stupid to try and cover his tracks, bringing hitmen down on his family. They go into witness protection...except for Tim, who actually is given a higher profile as Bruce's ward "Tim Drake" even though he's the Penguin's real target. Then he declares he's not going to be another Robin...but wears a carbon copy of the Robin costume and does the exact same job and incorporates it into his code name for no reason.

In story, he comes off as a reckless idiot who should have been sent underground with his parents, but wasn't.

And the out of world context doesn't really help. Lobdell wanted to make Tim more unique...by removing his discovering of Bruce's secret and making him a gymnast like Dick and a super-generic golden boy. Lobdell brings the parents back from the dead...but only so he can banish them again until he wants to use them as hostages. Lobdell is extremely professional when it comes to meeting editorial's demands...and somehow they feel they need to change his story without telling him here.

Seriously, the origin shows the weakness of having someone who's not a fan of the character tale a crack at him in a shifting continuity.