What happened when you changed the boot order and tryed to boot from the cloned drive? Any messages? Did you try moving the data cable from the c drive to the h drive (leave c disconnected) to see if it would boot?
The appearance of the 2TB drive looks like it may have been successfully cloned from the C drive. It's got the right partitions and the H partition is using about 200 GB (that's less than your 300GB or so on C, but temp folders etc are not copied so it possibly may be ok). If you look at H with windows explorer does it look like everything is there?
Regarding the change of motherboard. It's complicated. Part of the issue is the motherboard and disk drivers. But, there is also a dependency on whether the motherboard uses bios or EFI. If you backup or clone a system that was created on a bios motherboard and try to use it on an efi motherboard, it won't boot. Some drive backup/restore/clone programs try to address the driver issue, but I'm not aware of any that address the bios/efi issue.