My Internal HD won't boot
I'm assuming that the partition Backup OS X on your external drive is a bootable partition.

Turn on the Mac and hold down the option key. This will show you all the bootable partitions that the Mac can find. If the partition Internal HD in your internal drive does not show up then EFI has not discovered the required files for a boot partition and the OS on it can be considered damaged and should be replaced.

From what you describe I suspect some sort of damage to the low level formatting of the partition.

At this point I would boot from another partition and use Carbon Copy Cloner or a similar tool to backup the damaged partition and then erase and reformat the entire drive before installing a new System or a known good backup.