Sorry for the confusion. The links you are seeing each time are exactly the same, the difference in your results is due to time.
It takes a few seconds for us to edit a PDF document to apply the stamp, and the larger that file is the more time it takes and all of this has to happen in the short time between the last click of checkout and the next click to reach your thank you page.
In each of your tests you tried the link when it was shown in your browser immediately after checkout and stamping had not completed by that time. By the time you opened the email and followed it back to the very same page the stamping process had finished and you downloaded a stamped copy.
Under typical conditions we can have a stamped PDF ready immediately, but it looks like your files are a bit over 10 MB each -- that's not too large, but it could be adding a few seconds to the process which seems to be making all the difference.
If you wish it would be possible to set things up so that the email is the only way to get the download links, so that buyers won't be able to start their downloads before the stamping process is finished. Follow these steps:
1 - Create your own static thank you page, just a simple page on your site that confirms the order and instructs the buyer to wait for an email with their download info. It's a good idea to include your contact information on this page in case they have problems or questions.
2 - Go to Seller Admin > Edit Account Preferences and enter the address for that page in the Common Thank You Page field
3 - On the same screen, erase the entire text for your Common Thank You Email. The reason for this is that the Common email only links to your Common thank you page, and you've just replaced that. There's no need for buyers to get an email that sends them right back there again.
4 - Make sure your product emails are still working. They are on by default so you probably don't need to make changes, but just in case go to Seller Admin > View/Edit/Delete Products and Edit your items. Make sure the email settings for each one are in one of the following configurations:
* The email field is blank and the "Enable Templated Email" box is NOT checked, which means we send a default message including a working link for downloads. (default settings)
* The email field has your own short custom message and the "Enable Templated Email" box is NOT checked, which means your message will be sent along with our default message and link
* The email field has your own detailed custom message that includes the [%download_link%] template AND "Enable Templated Email" IS checked. We will not send our default message, but the download link will be contained in your message that is sent out.
With these changes in place the buyer will checkout, reach your page which instructs them to check their email. The email should arrive immediately, but the extra time involved in checking the email and following it to the download page will help make sure the first download does not happen before we finish stamping the file.