Drug or Cosmetic
Here's where it gets really confusing: Whether or not a product lists the active as an active ingredient on the label comes down to whether or not it's considered a cosmetic or a drug. A skin-care cosmetic product may contain the same active ingredient as a product that's considered a drug, but it won't list the active ingredient in question as an active ingredient on the label, but rather as an ingredient even if it is advertised on the front of the packaging. This is due to the wording of the claims around the ingredient. For instance, a face wash containing salicylic acid may be considered a drug and list it as an active ingredient if it claims to actually treat or manage acne. If it's cosmetic, it may just list salicylic acid among its many ingredients.
It's the way a product is marketed including its intended use, what the consumer perceives it to do, and what the packaging claims it will do. That determines which category the FDA puts it in.
So, a product might say it can "reduce the appearance of" skin concerns such as wrinkles or redness or otherwise make them "less noticeable" without specifically saying it treats the underlying condition associated with those issues. In these cases, the FDA treats them like cosmetics rather than drugs. Cosmetic ingredients aren't tested by the FDA before they're sold, so the responsibility for them to be safe and effective rests on the manufacturer. Specifically, cosmetics are defined as products "for cleansing, beautifying, promoting attractiveness, or altering the appearance," and they are required not to mislead consumers with their claims, otherwise the FDA may take action.
In 2016, the FDA issued warning letters to 30 different companies for citing drug claims associated with topical skin care, hair care, and eyelash/eyebrow preparations, noted on both product labeling and websites. Some examples of the drug claims cited are acne treatment, cellulite reduction, stretch mark reduction, wrinkle removal, dandruff treatment, hair restoration, and eyelash growth. All were selling a product as a cosmetic but labeling and promoting it as a drug, e.g. intended to diagnose, mitigate, treat, or prevent disease, or to affect the structure or function of the body.
Bottom line: be a label and ingredient reader! Be wary of companies claiming to cure a health concern. See if the label states the product is FDA approved. Check with your doctor if you are unsure of the what the product claims.