Maybe you are confused because there are so many different products, e.g., no splash, regular, non chlorine, etc. I think I saw at least 7 different formulations all called Chlorox.
Regular bleach is what everyone is talking about, not some substitute that is intended to make colors brighter. Regular bleach is Sodium hypochlorite which ruins cottons by turning them yellow. So there are other "bleaches" that contain other substances including enzymes for special purposes.
If you are smart enough to know the word "halogen" then you should be smart enough to know what chlorine bleach is. There may be some products with bromine or iodine, but I wouldn't want to use them on my clothes. And certainly, none have fluorine in them.