How do you find out what payment processor(s) a given site uses?
I know that some provide methods whereby a site can have the actual payment entry form served and processed by the payment processor instead of by the site's own server, so you'd be able to see from the user's end where they payment is actually being processed.
I've never done a survey, but just anecdotally most sites I've encountered seem to not be using that option. Their payment entry form comes from their own site and posts back to it, where their own back-end handles dealing with their payment processor's API.
Using the method where the user interacts directly with the payment processor does have the advantage that it simplifies PCI compliance. If your systems never even see the credit card, just receiving a token from the payment processor at the end of the transaction that you can use to initiate subsequent on-file or recurring transactions, most of PCI goes away for you.
On the other hand, that also means that you are stuck with that payment processor for on-file or recurring transactions for that customer. Your token from payment processor X is completely worthless for doing charges at payment processor Y.
If I was in a business that has a significantly above average risk of running into payment processor trouble so I might need to change processors, I'd want to store the credit cards myself. That makes it possible to change payment processors without having to get all of your subscription customers to come back and re-enter credit card information [1].
[1] Well...at least for now. I'm not sure if that will still be possible if the Visa stored credential framework ever actually becomes required. Briefly, under the SCF requirements when you store a credit card, you have to send a flag to Visa with the transaction saying you are storing it. On subsequent on-file or recurring transactions, you have to send a reference to the transaction that stored the card.
The problem is that you reference that transaction by sending Visa's transaction number. But Visa's number for transactions is generally not the transaction number you get from your payment processor. The payment processor has its own transaction numbers and those are what you see.
I believe MC is also doing SCF. Not sure about Discover and Amex. It was supposed to become mandatory something like two or three years ago, but payment processors kept asking for extensions.