from: http://www.textileaffairs.com/stains.htm
It says:
Household liquid bleach, identified by the words "sodium hypochlorite" in the ingredient listing is the only bleach that disinfects. It is also the most effective bleach for stain removal and whitening. This is the bleach of choice for whites and colorfast garments.
...
Clorox BottleColor-Safe Bleach is, quite simply, a bleach that is safe to use on washable, colored clothes. These bleaches, available in powder and liquids, remove stains and help keep colors bright. The powdered bleaches also have enzymes to break up tough protein stains, which is why they make an excellent presoak.
* A color-safe bleach is not a diluted bleach. The gentle bleaching action in liquids is from hydrogen peroxide. Powders commonly use sodium perborate.
* These color-safe bleaches also have optical brighteners. This ingredient is like an invisible layer on fabrics. It reflects light, making colors appear brighter and whites whiter.
Wikipedia says Sodium hypochlorite is NaClO, and thus does, indeed have chlorine in it. It indicates it is the chlorine based acid which provides the cleaning action.