My payment processors (Shopify, PayPal, Klarna, etc.) don't treat debit cards any differently than a credit card. Those are all huge payment processors; so it's not like a special snowflake here.
Since we're on HN, I'll mention Stripe. I don't use Stripe - but they act the same way.
All of these don't bill for "credit card" they bill for "card".
Because it is their choice.
> I'm happy to offer a discount if someone wants to pay via a non-credit card, but I don't raise my prices because of it
This makes no sense to me. As the leader of the business, you have the option to
1) deny all credit cards
2) accept all credit cards, but not offer discount for debit cards
3) accept all credit cards, but offer a discount for debit cards
It is just a business decision, and if you are not choosing 3, one would hope it is because you are earning more profit by going with option 2.