Here's my updated speculation (updates in bold):
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. Moreover, the Legend engine was already running on several platforms and a remake based on that engine and published on multiple platforms would be more lucrative. It also explains why Crystal Dynamics doesn't necessarily have PSP expertise in house; they're designing and Buzz Monkey Software is already under contract to handle a PSP release. (What is unclear at this point is why the Xbox and Xbox 360 are not included, as they also have ports of the Legend engine.)As always, I'm open to suggestions and extra information.
For maximum effect, SCi/Eidos conspicuously didn't announce anything at E3, and then started some well-chosen leaks afterward. Crystal Dynamics were the first anonymous tip to Kotaku.
Core felt spurned. They knew they were being sold to Rebellion and that Crystal Dynamics had to be working on the same game, because the Kotaku leak mentioned Tomb Raider II, which isn't in their version. (Everything I've seen in the Core-produced trailer is material remade from just the original Tomb Raider -- nothing is shown of Tomb Raider II.) Moreover, some may have felt a personal enmity toward Toby Gard for working directly against the interests of his former company. So someone at Core decided to stick it to SCi/Eidos and Crystal Dynamics by giving the public its first taste of a remake -- Core's remake -- along with the ominous and misleading "cancelled" project information.
SCi/Eidos tried to control the damage quietly, but were forced to put out their press release and pressed Core to disavow the trailer release. They also broadened the target platforms, hoping this new information would blunt disappointment some might feel at losing the Core version of the game.
Behind the scenes Eidos had made a business decision: fan blowback and faithfulness to the original were less important than a multiplatform strategy utilizing the tagline "...from the creators of the hit game Tomb Raider: Legend".
The last anonymous tip to Kotaku is the most interesting. While it still takes the line that Tomb Raider and Tomb Raider II are being put into the remake, the tone at the end indicates to me that the Core trailer looked very good compared to whatever Crystal Dynamics has put together. I think this last tip came from a well-placed journalist who has already previewed what Crystal Dynamics has done and compared his/her notes with what all can see in the trailer.
If Crystal Dynamics has come up with something that isn't as impressive to fans as the leaked trailer from Core, it will be interesting indeed to see how well it is received. My gut feeling right now is that fans will be disappointed at the comparison, but SCi/Eidos will still be pleased with the profit margin from a high-profile multiplatform release.
Finally, I have now read on a forum that members of Core made public forum posts claiming their game was also multiplatform, hitting the PSP and PS2 and Windows. Until there is something more substantial, I'm not including that information in my speculation.
How do you know Core wasn't the source of the last anonymous tip? Who else would want to bash CD like that while praising Core?