Working for nonprofits and church's, I know we do it because AMEX and others charge higher rates per transaction than say VISA, thus some clients only take VISA and giving a select list of options, lets the user know to use that type of card.
Not taking Amex for cost reasons is almost always a bad move. First, it annoys the person giving you money (always a bad move). Second, Amex cardholder spend way more than average (frequently a bad move). They may still spend as much when asked to use Visa instead, but maybe not.
I'm an AMEX cardholder. I don't really mind too much when people don't take AMEX. I understand they are a "premium" brand and I adjust my expectations accordingly. That being said 99.9% of the time they take AMEX.
>giving a select list of options, lets the user know to use that type of card
No. AMEX cardholders know they aren't welcome 100% of the time, and they know to look for an AMEX logo before typing in their info. You need to have the proper logos or lists of payment methods right next to your way to pay.
Seconded. I know to look for an AMEX or Discover logo at a cash register or on the bill cover at a restaurant/bar. Those cards are accepted at many more businesses than they were ten years ago, but acceptance will never hit 100% due to the fee differences for merchants.
We do stress that point, that the barrier to donate should be as low as possible. Still, some churches go as far to say donations can't/shouldn't be made on credit do to their individual beliefs, and would only take ACH transfers as they pull directly from a bank account.
Merchants are now seeing a lot of payments by debit card which are not loans (direclty out of bank account). And there are still a lot of Amex cards out there that require full payment each month which is not really a loan.