This is not true. Anyone in the US who has told you this is confused or getting ripped off by a predatory ISO (the companies that offer credit card processing).
Numbers below are only looking at the percentage, not the flat per txn fee or the other fees like assessments, etc... Only apply to the US. Also assuming you do less than 1 million a year in card volume.
First let's look at actual merchant services costs:
Stripe, Braintree/PayPal (online) you pay the same fee for all cards. Around 2.90%
Wells Fargo (in-person) charges the same rate for all cards until you are high volume. Around 2.40%
Chase (in-person) charges the same rate for all cards until you are high volume. Around 2.60%
Bank of America (in-person) charges the same rate for all cards until you are high volume. Around 2.65%
Now let's look at the actual amount the card issuer charges for a restaurant to accept a card (assuming you're working with an ISO that does OptBlue, all the above do). The markup you pay your merchant services provider is added on top of this.
The difference is not that high. In fact, my understanding is “Visa Signature” and “World Mastercard” (which are most of the premium credit cards) normally cost about the same as Amex. At worst it may be about 1% more not double.
"payment cards" includes debit cards, credit cards, and other types of cards that can be used for payments, such as the German girocard standard. There's no meaningful difference between these products if your locale has working consumer protection laws.
The payment processors I'd be looking at offer merchants rates of 0.125% for girocard, 1.39% for VISA/MasterCard, and 6.2% for AMEX.
High fees are what Apple uses to offer such “great benefits” with their existing card. If they want to expand to Europe, the only cards enabling that would be Amex, or Diners (which is even less accepted I think).
visa and mastercard are 2-3%, amex is 5-6%. there's no upside for merchants to accept transactions where they have to pay double the processing fees