In LN, how could a payer prove to 3rd party that he indeedly made a payment to the payee? For regular on-chain payments, a txid can prove at least a transferring of value indeedly happened, at a specific time, despite that it may be totally irrelevant. For example, a malicious user can send his own money to himself, to pretend that he purchased some goods from an innocent merchant.

For LN off-chain payments, is it possible for a payer to prove to 3rd party that he had indeedly made a payment to the payee?
In lightning the receiver of the money comes up with a random secret and comits to it via the payment hash. Once all the hashed time lock contracts (htlcs) are set up along a path through the network from the sender to the receiver the recipient will release the secret preimage for the chain of htlcs to settle. The preimage can be seen as a proof of payment. However as with the onchain case there is some doubt. Every node along the route who participated in the payment will know the primage of the payment hash and could thus argue that they initiate the payment