Hi,
Regarding these settings, the badge should be displayed only on the product "Dermatologica Pre Cleanse Cleanser" if this product is in one of the categories selected under and if these categories are selected in the discount "10% off Dermatologica".
So I think that you should be less restrictive in the rules for this badge, you should only select the desired discount (which can be limited to some categories) or only on the desired categories, or only on the desired product, but not fill all the field for one badge.