"For reasons unknown, Core and Crystal Dynamics were working independently on the same 10th anniversary game. Perhaps Core had been given a green light to try and after the SCi/Eidos merger, no one remembered that detail. Meanwhile, Crystal Dynamics successfully launched Tomb Raider: Legend and was considered by the management, especially in North America, to be the obvious team for the new game."
We've embedded the video for Core's version after the break. If anyone can track down the Crystal Dynamics video, we'll post that too and we can tear ourselves apart arguing over which one would have -- undoubtedly -- been the better game.
[Update: Curmudgeon Gamer has updated their analysis writing, "According to the SCi/Eidos press release verifying that Crystal Dynamics would produce the game, it will be published not just for the PlayStation Portable but also for Windows and the PlayStation 2. The current Tomb Raider: Legend engine supports all three of these. That changes my reading of the situation."]