To be honest I'm not really sure. I can only presume they are implying they are on the same exchange. I've only just noticed that particular turn of phrase in the last four to six weeks perhaps. A lender only has to start a scam accusation and a genuine borrower will go to any length to prove they had made the repayment.
Yeah, lately exchanges have the internal transfer feature if the spending of coins is done using addresses from the same exchange. They won't bother transferring the coins on-chain but will instead just credit the account internally/off-chain
In such cases, there will be no on-chain TX ID which helps prove if the transaction was actually made using blockchain explorers. I agree with you. In case a dispute comes up, getting evidence about the transfer can be quite a hustle.