Debiting the bank in accounting system in the credit note to customer where usually is Accounts Receivable account.

You can debit the bank or use no. 1 bank asset account in the credit note issued to customers for double payment as per first picture (asset bank account code 1-1110 is used instead of 1-1200 Trade Debtors account which is usually used for credit notes). 

Why is the bank account used in this case?

It is because we want to reconcile the bank before we know what the customers want to do with this double payment. Option 1) they want refund Option 2) they want to keep money with us and apply this credit note against their future invoices. 

We have received a payment for an invoice which is already paid last financial year in this case. If the invoice they double paid is in this financial year, we can apply double payment against the same invoice and making overpayment credit note itself in that invoice, meaning they overpay this invoice by doubling it. However, in this case, the double payment relates to previous financial year invoice and the accounting software does not allow to make changes to already closed/paid invoices in previous financial year. Thus, we have to issue a credit note separately as below. The head of department wants bank reconciliations to be done on a regular basis. Thus, we have to accommodate that/reconcile the bank, thus leads us to use the bank account code instead of Trade Debtors account which we usually use. In this case, by debiting/using the bank account code below, we are making -$4,626.59 less in that bank account. Thus, bank reconciles. 

In this case, the customer confirms that they want to keep the money with us and use this credit note against their future invoice# 14899. In the second screenshot, we issued the client this invoice# 14899. You can see the accounting journal entry for this effect as well there. In the last screenshot, the first credit note# 14844 was applied against their future invoice# 14899. The accounting logic is if you see all there 3 journals, we can see that the accounting equation is achieved. Trade debtors cancelled each other out. What is left is below

Dr NAB cheque a/c $4,626.59

                                                  Cr Project sales income a/c  $4,205.99

                                                              Cr GST collected     $420.60 (tax Aus specific)