Adrian:
The basic problem is that you flashed a ROM not meant for your model of the S3. Once you've done that a factory reset won't help, because all it does it try to reinstall the same ROM you flashed. Like I said above, all S3s are
not the same. You should only flash a ROM that you know is designed for your exact model and carrier of the phone.
After you rooted, but before you flashed the new ROM, did you make a nandroid backup? If so, try to restore that from recovery.
Again, it's really hard to see what's on your phone screen; it's blurry, and you keep moving the phone around. But I think you were able to boot into recovery. If that is correct, you should be able to fix this by flashing a ROM that will work with your phone. Got to the forum dedicated to the i9300:
http://forums.androidcentral.com/samsung-galaxy-siii-i9300/ , tell them the exact model # (look on the box, or take the back off and see what it says under the battery), and what country and carrier you're on. Someone should be able to point you to a good ROM.
If you can't flash a good ROM in recovery, then you should be able to load the original factory ROM using Odin (download mode). Again, people in the forum above should be able to point you to the right ROM once they know your exact model, country and carrier. In the future, ask if the ROM you want to flash will work on your phone BEFORE flashing. And always make a nandroid backup before flashing a new ROM.