To best keep track of whether an invoice has been paid or not, you will want to associate a payment to it.

You can do exactly that by assigning payments against invoices. This is what the 'Status' badge tells you, it'll show you whether an invoice is Paid, Not Paid, or Partially Paid.

To assign a payment, view the invoice then select the 'Assign a payment' button

Next, you can choose an existing payment already recorded (if exists), or record a new payment on the spot.

Once the payment has been applied, the invoice paid status will be updated to reflect that the invoice has been paid.