As the issue is whether one has to pay out a lot of money with the prospect of a lengthy and time-consuming fight with LH (likely to require an MCOL claim), I would call LH back and if I received the same answer, ask to speak with a manager and make clear that I had not cancelled the ticket.
From the face of the record, it is likely that the agent who performed the transaction did indeed cancel and refund the ticket and that is all the front line agent sees now.