What are the specs of the HDD you're having problems with?
That 800283F0 code looks awfully familiar to me as I ran into this same issue when upgrading my HDD. The built-in backup and restore utility on the PS3 is known to be rife with problems, although the consensus seems to be that it's caused more by the actual game data itself being corrupted, and not an issue with the utility program. The more data you are trying to back up and restore, the greater the chance you'll receive an error -- unless you can narrow down exactly which game data files are to blame.
Something to try: put your old HDD back into your PS3 (assuming you haven't deleted anything or formatted it); it should start up like normal. Delete all non-essential game data and patches. It's not necessary, but I highly recommend backing up your individual game save files onto a smaller USB thumbdrive (for those that aren't copy-protected). Then run the full backup utility again. If that means you'll be deleting everything, then it's kind of a moot point to use the restore function.
Anything you've downloaded under your PSN ID can be re-downloaded through the PSN Store Download History.
FWIW, I've read that 2.5" HDDs larger than 640gb tend to have more problems with error codes and system freezes compared to those at 640gb or less. I remember seeing a compatibility list published by Sony a few years ago, but they must have scrapped it because the official Support site now states that they "[do] not have any information regarding specific compatibility or recommendations for hard disk drives for the PS3 system. Proper operation is not guaranteed for all models."
