The Free Checkout page is an aspect of our system shared in common among all sellers using our service, so it cannot be customized, aside from uploading a custom logo/header image (in Seller Admin > Edit Profile).
The Payment Processor column only indicates the checkout method (PayPal, Google Checkout, E-junkie Free Checkout, etc.). We only indicate the Discount Code(s) applied to the order and/or show "Current Promotion" if the order was eligible for an automatic (no-code) discount; the Discount Name and amount are not shown in the log. Discount Codes are also shown for each discount in your Seller Admin > Cart Discounts > Existing Discounts menu. You may wish to use Discount Codes that all have a common prefix (perhaps corresponding to the Discount Name or indicating the partner distributing your codes), so you could identify the specific discount applied that way.