If I understand you correctly you want to run your OS from the external HD?
this can be done -
Please note that it will be slower if HD is connected over USB, faster over ThunderBolt and always best and fastest from the internal SATA hard drive.
If you want to continue and install OS onto external HD
please restart the Mac and hold CMD+R keys.
This will take you to recovery mode.
Please use Disk Utility to partition and format the new drive.
Run recovery installation by pointing to the external HD.
Once you are done you can boot your computer either to your original internal drive or your new installed external one. You can choose it in Startup Disk in System Preferences.
Post if you have more detailed questions.
Please note, I've requested this thread will be moved to the Mountain Lion forum. 10.3 has nothing to do with Mountain Lion, as no computer compatible with 10.3 is compatible with Mountain Lion. No action is needed on your part as far as reposting. This will hopefully also get you some more responses.
Another way is to simply clone your current startup drive to the new drive. Make sure the new drive's partition map is GUID. You do this in Disk Utility when you partition, or repartition the drive. You can also check a drive's map in Disk Utility. Launch the app and highlight the 2 TB drive's physical drive icon. Like below:
Then look at the lower right of DU's window. If the drive is already GUID, it will say so, as per the next example:
Once the new drive is setup correctly and ready to use, you can clone your current startup drive to the new one using Disk Utility, or one of two excellent cloning utilities; Carbon Copy Cloner, or SuperDuper!. This will save you the trouble of installing OS X and incorporating your Time Machine backup to finish. It will already be done as the new drive will be an exact duplicate of your current startup disk.