The correct answer is that when you install FFXI to your 360HD, the game is also license bound to your specific console. This is done specifically to prevent you from moving your HD to another console and continue playing (god knows why.)
The solution is to go to your system settings > memory > marketplace > and delete the download cache. This will also delete any game updates you've downloaded from the internet.
Then go to xbox.com and log in with your email and password. Search for "license" - the first result will walk you through downloading a new licence for your new system with the old harddrive. The next time you connect to xboxlive on your new console, a download will begin automatically. Problem solved.
Also, slightly related, the install for FFXI does not show up on the memory section of your system tab. I assume that it's to prevent you from accidentally deleting it. It can only be uninstalled from the install discs. This is why your HD is probably missing ~6gb of space that you can't account for.
For the love of god people.
LOOSE is an adjective, LOSE is a verb. LOOSE is an adjective, LOSE is a verb. LOOSE is an adjective, LOSE is a verb. LOOSE is an adjective, LOSE is a verb.
Please stop saying you loose things. (besides arrows, of course.)