As far as I can tell the messages were through the dispute system as the subject included the dispute number and included the text:

 "The seller has responded to you and offered another solution to solve this issue."

The messages are logged by eBay under the dispute.

All of that said I was not really aware that there's more than one way to exchange messages between the seller and buyer. The seller did make requests that "I wait a week longer" and "Check with my local post office" so it could well be that the seller has engaged in some very dodgy behavior knowing full well how the system works. On the other hand whilst I realise that eBay needs to close disputes in a timely fashion I find it remarkable that I wasn't asked if the matter had been resolved before automatically closing the dispute.