I have worked on and managed a lot of e-commerce sites. If you can open a merchant account with a bank (which is not necessarily easy) you can use any number of processors. Braintree is the one several of my clients use. PayPal owns Braintree but it works separately from PayPal’s credit card offerings.
If you don’t have a merchant bank account you have to use an intermediary like Stripe or PayPal. They use their own merchant account and charge higher fees. I’ve used both and find them about the same, though I think Stripe integration with a web site is a bit simpler. Most of my clients use Stripe and none of them has had serious issues. When I’ve had support issues I got them resolved quickly, though it seems like PayPal has more bureaucracy.