The reason for offering a discount for paying in cash is probably that most clients would get angry if merchants tried to charge them more for using a credit card while they are always happy when they get a discount for paying in cash. I think the credit card processors are smart enough to realise that this is happening, so if they wanted & could do anything about it, they would probably write their terms differently (cca. "prices for payments in cash and with cards must be equal").

In the UK, a majority of bars and corner shops will have a 50p charge for card use below a certain threshold or even refuse to take payment.

In the past I've just offered hesitant merchants 50 cents extra. Probably only works when the merchant also owns the business though.

They do, but they shouldn't. Both MasterCard and VISA explicitly prohibit this practice in their terms.

Plenty of places demand a minimum payment for credit card transactions, with an additional fee in lieu of that.

If they do, they aren't enforcing it at all in new Zealand, where a 2% surchage for credit cards is fast becoming the norm.

Cash can often be used to fiddle the accounts too ... which also leads to a cash preference for some.

That used to be the case, but as stated by the above poster, it is no longer the case due to a lawsuit a few years ago.