I need to send money from my paypal account either to a bank card linked to a PayPal, or directly to a bank card, not necessarily linked in a PayPal account. I have used some API's that send money within paypal accounts which work perfectly fine, but now having the feature to send directly to bank cards would be even better.
I know that when you pay using a bank card and then do refund, the money goes back to the bank passing over the paypal account, which is great, so there should probably be a way that paypal can transfer money to bank cards using Paypal API.
PayPal send money APIs such as Payouts can be used to send money to PayPal accounts or email addresses that do not yet have a PayPal account. If there is no PayPal account, the recipient will receive a message and have 30 days to create one to claim the payment, otherwise it will be automatically refunded.
Once the recipient receives the money into their PayPal account, they can withdraw it to any compatible place for their account's country -- including bank accounts, and cards that support withdrawals.
To use PayPal Payouts, the account sending the payout must be approved by PayPal to have the service enabled.
It sounds like you've already made use of a service like the above.
For large enterprises and marketplace platforms, there is a hyperwallet service by PayPal. You can see a feature comparison here: https://www.hyperwallet.com/features/ , but it is not likely to be an option for your case.