That agrees with the WRF entry for cleanser in #3 above - i.e. a pretty broad rangeOr even 'cleansing agent'.
Cleansing agent - definition of cleansing agent by The Free Dictionary
cleansing agent - a preparation used in cleaning something
If you add the single word cleanser, it seems to suggest that it is usually used without (those) adjectives, and presumably the context would be enough. If we had any from the OPCleanser is the word used by advertising agencies to refer to whatever cleaning compound they're trying to sell. An early advert for Ajax scouring powder was "Ajax, the foaming cleanser!" (See Ajax (cleaning product) - Wikipedia, the free encyclopedia.) You wouldn't use that on your skin!
By and large, it's only the cosmetic products that are called simply 'cleanser' in everyday English.
So I see I'm using the same evidence as Parla in #11 to come to the exact opposite conclusion!
A comparison of the various collocations as in Google Ngram Viewer is interesting, but proves little.