I am in the USA, and it looks like that site is based in England, so the way they divide the breeds is a bit different than what I am used to seeing. So I am just reading what the pages say and trying to make sense of it.
The difference between "Old English Bantam" and "Oxford Old English Game"...
I never heard of them before you asked your question, but then I found a website with a few photos:
https://www.poultryclub.org/breeds/chickens/hard-feather/
They have both a Carlisle Old English Game and an Oxford Old English Game, along with a few other kinds of "game" on that page.
They...