The Stamper only applies the buyer's name, email address and unique Transaction ID to each page of the PDF they receive.
As explained on our PDF Stamping help page:
4. Ordinarily, if you update a product's file and the buyer uses their original, unexpired download link again after that, they would get the latest file you'd uploaded for that product; however, if you enable PDF Stamping, each buyer's original download link will always issue their original Stamped file, even if you re-activate their link;
5. Free download links issued via the Updates function in your Seller Admin cannot issue Stamped files;
6. New! Links issued manually via the Send Free Download Link function in your Seller Admin now can issue a Stamped file.
If it would be too cumbersome to issue each previous buyer a new link manually using Send Free Download Link, you might consider creating a product-specific discount code good for the full purchase price of that product, perhaps with an expiration date a week or month hence (do not set a Max. Use Count, as that only limits how many times a discount can ever be used, not a per-customer use limit). Then you could use our Newsletters service to send that code to previous buyers inviting them to use that code to order a new copy for free: