This happened to me also, took awhile to figure out. Game is the breed, just like Wyandotte or Sussex are breeds. It's a very old breed, so it's confusing. There is an American Game and an Old English Game (OEG). The other names you are seeing are 'strains' or 'lines' but they don't matter unless you intend to show them at competitions. They are all the same breed. If you'd like to know what strain you have, there is a great thread for that in the Gamefowl area.
So-
Species: Chicken
Breed: Gamefowl
Strain: Kelso or Sweater or Roundhead or other...
Line: Jimmy John Joe or Mick Smith or other...
Also, this is totally, completely different from 'Gamebirds' like Turkey or Pheasants, which are different species, with their own various breeds.