Can I Use an Older 360 Hard Drive as my Primary Storage Device through a Transfer Cable?

I also wanted to mention that if you did attempt to use the transfer cable in order to use the storage device that you would end up only having access to 32 GB of space from your hard drive as it would be considered an external device.

nope transfer cable is only for transferring data from one console's hard drive to another. you would need to get a 250 a 320 or a 500 gig drive. then plug the cable into the new 360 and plug your hard drive onto the other and and transfer it to the new
system. Unless your saves will fit into the 4 gig drive (you had the size wrong its a 4 gig model not a 5 gig model).

When you transferred the content over to the new hard drive, did you have the old or new hard drive plugged into the transfer cable?

If you had the 120GB hard drive plugged into the transfer cable that may be where the issue is being caused, would you be able to transfer the data back to the 20GB hard drive, plug the 120GB hard drive into your console, format it then try transferring
the data back to the 120GB hard drive with the 20GB hard drive plugged into the transfer cable and the 120GB hard drive plugged into the console.

Scroll down and follow the "Format a storage device":
support.xbox.com/.../storage

Let us know how it goes!

Are these the steps you tried?

Copy or move multiple items using the Transfer Content option

You can use the Transfer Content option to move multiple items between storage devices. Here's how.

Notes

• The Xbox Dashboard is not available during content transfer.
• If the content transfer is interrupted for any reason, the content remains on the source storage device, and you can restart the content transfer process at any time.
  
• If you’ve purchased a new Xbox 360 console, you can use the Transfer Content option and the Xbox 360 Hard Drive Transfer Cable to move your Xbox 360 content to your new console. For more information, see
  
  Use the Xbox 360 Hard Drive Transfer Cable.

1. From Xbox Home, go to settings, and then select System.
   
2. Select Storage.
3. Highlight the storage device that contains the content you want to transfer, and then press
   Y for Device Options.
4. Select Transfer Content.

Yes, you can still use the older transfer cables to transfer from the older style hard drives to the newer hard drives. As long as it's an official MS cable you shouldn't have any issues.