Let's suppose that you want to buy something Online using bitcoin. Is it provable (to a 3rd party) that you payed using:

1. Bitcoin on-chain - this may seem straight forward because the transaction is on the blockchain and everybody can verify. However, in a dispute, the merchant may claim he didn't receive the payment and that the address I am referring to in the dispute is not his.
2. Bitcoin lightning network - here I don't know if there is a way at all to prove the transaction to a 3rd party.

For on-chain, I am thinking of a payment request that includes a signature that everyone can verify. The public key could be included in a DNS record (TXT record for example) of that domain. For example if I am buying something from the site abc.com, this domain should have a TXT record with the public key of the merchant. When I pay, the merchant will show me a payment request that is signed with his private key. I verify the request using his public key, then save the request and pay (all these steps can be done automatically by the wallet). Later in a dispute I can prove to anyone by providing the signed request, that the request originated from the merchant and that the address is his. The transaction itself can be verified on the blockchain.

For lightning network, I have no idea if a similar approach is doable.

What do you think about that?