This a question about places (such as pubs) that takes all the normal methods of payment, but also have a big sign up saying something like “cash preferred”. I had a look in taginfo and cash=preferred was there, but at a very low usage.
Is there some more popular tag that I’ve missed, or is that concept just not recorded much?
Three not a lot. Also payment:cash=only usage looks low. Prompted me to update a local pub with that, found it has payment:credit_cards=no and payment:debit_cards=no. I do not think *=unpreferred is used much more on those means of payment.
payment:cash=only
payment:cards=yes
payment:cheque=yes
That seems to be an odd meaning for only? The wiki page mentions payment:cash=yes; payment:others=no but doesn’t say anything about only not really meaning “only”. All the ones I can see in UK/IE with payment:cash=only and payment:cards are payment:cards=no.
It was a tongue-in-cheek, which I tried to indicate by the emoji…
While it is a form of trolltaging, it does make some sense if you interpret it as “we only take cash, but will reluctantly accept other means”. So, unless you want to venture with preferred (or, god forbid, designated), perhaps it’s not so preposterous. 
preferred sounds reasonable for the OP case. designated would be indeed more appropriate (to conform with other keys that have the same value for the “very obvious” meaning of it 
).
The interpretation of the three aforementioned tags sounds more vague and confusing than designated tbh.
my vote for cash=preferred. designated may be more in line with the same value for other keys but does not mean the same as preferred.