FDA recently updated the web page, “Information for Cosmetic Importers.”  This document uses a question and answer format and the update offers some clarification on the use of INCI (International Nomenclature Cosmetic Ingredient) names for cosmetic ingredients:
Is INCI nomenclature acceptable for identifying botanical ingredients?
INCI (International Nomenclature Cosmetic Ingredient) names for botanical ingredients typically use Latin names for genus and species to identify botanical ingredients. U.S. regulations require the use of common or usual names. Latin terms may be included parenthetically following the common or usual name of an ingredient. Example: Aloe (Aloe Barbadensis) Extract. See “FDA Response to CTFA Requests Regarding Harmonization of Ingredient Names (Color Additives, Denatured Alcohol, and Plant Extracts).”
It is very helpful to have this information spelled out by the agency since previously I and others have received conflicting information from the agency on the use of INCI names.
The related topic of C.I. numbers (used in European labeling) is also covered in the update:
Are C.I. numbers acceptable for identifying color additives on cosmetic labeling?

C.I. numbers are not acceptable on product labeling unless they are preceded by the color additive names accepted in the U.S. followed by the C.I. number in parentheses. In addition, C.I. numbers do not indicate FDA approval or FDA color certification. To learn more, see “Color Additives and Cosmetics” and the additional resources listed on that page.

Unfortunately, FDA has not been able to provide true clarity for what ingredients are and are not allowed in cosmetic ingredients.  The import update states that not all ingredients permitted use in cosmetics sold outside the US are permitted in the US and links to a very short list of prohibited ingredients.  But the update goes on to state, “But remember, any ingredient is prohibited if it causes the finished cosmetic product to be unsafe for consumers under labeled or customary conditions for use, even if there is no regulation specifically prohibiting or restricting its use in cosmetics.”  The associated link leads the reader to the brief list of prohibited ingredients and this Q&A:
Are these the only ingredients that can cause a product to become violative?
No. A product may become adulterated or misbranded in a number of ways. Among these are improperly used color additives or any ingredient, other than a coal-tar hair dye, that causes a cosmetic to be harmful under customary or usual conditions of use, or cause a cosmetic to become misbranded, as stated in the Federal Food, Drug, and Cosmetic Act, Sections 601 and 602. Additionally, an ingredient having a history of use in or as a cosmetic may at any time have its safety brought into question by new information.