Customer Center Role - Invoice Sales Order turned out to be showing as Cash Sale

When configure customer center role, for the Transaction Forms for sales order, only forms available for me to configure is Sales Order (External) type. None of the regularly used sales order forms are available for me to assign to a customer role.

I picked a external sales order default form "Standard Online Order - Invoice", customized it. Customer is able to see that form and create a sales order using that form However, even though this order is Invoice type, when I login to backend and trying to edit this sales order created from customer center, it is showing under a Cash sale form that I customized for internal use (I have one invoice type of sales order form, one cash sale type of sales order form, the cash sale form is a preferred default one).