Is it really adding another payment method? I thought it was me > p2 seller > me. Not me > p2 seller > Binance > me. Meaning the seller is the one who is sending the money from their account to the account I give them. It's not Binance that's sending me the money from their account.
You are missing the point. You are still using Binance to perform the p2p transaction, right?
This means in order for you to add the payment method, it has to match with the name you used to verify the account.
Secondly, there is no sane buyer who is going to allow sending money to a third party whose names do not match your payment options available. This is how people get scammed.
If he sends money to a third party, it means if you don't release the bitcoins and he tries to appeal. It means he will lose the appeal since the person he made payment to has no details listed in your payment options... basically, he will have no evidence to show that he paid to your listed payment methods.