I own a CF gym.
@paul is correct. CrossFit with the capital F is a trademark. It cannot be endorsed or marketed unless the trainer is L1 certified. A gym cannot promote classes unless it has an affilation with CF.
Cross training is a type of training and can never be trademarked.
Heres my gyms website: http://www.crossfitnaples.com
CrossFit is a brand name and/ or a company that promotes a specific exercise philosophy and holds competitions.
Cross training is simply the practice of using several different modes of exercise.
Cross training is good for preventing repeated use injuries. You might know someone who ONLY runs, or ONLY swims, or ONLY cycles/rides bikes. These people can often times become injured from doing the same repeated motion over and over instead of mixing in different modes of exercise.
From what I understand, CrossFit seems to promote cross training in its programs.