Even when the brand is as big as PayPal, I think that the OP's point stands. When you "Pay with PayPal," you are generally using your PayPal account. You don't need a Stripe account to pay, so you're not paying "with Stripe." PayPal is the system, Stripe is an implementation detail.

Hear, hear. I actually wrote just that at first but dropped yet by mistake when I reworded my post. Not yet, but here's hoping.