How do I mark an invoice as paid?some invoices are marked 'paid' and the same invoices marked as 'closed'. is this double counting? which is correct?

There is no double counting, brian-scally.

The ones that are showing as Paid are the invoices and the ones that are marked as Closed are their Payments. If you notice, invoice amounts are showing positive figures which add up to the customer’s balance. While Payments are negative figures which offset the invoices. 

There isn’t a way to mark payments as paid. By default, it is marked as closed when completed. Also, deleting a payment can make the invoice status open. 

Please feel comfortable on posting here again if you have further questions.