If I have a desktop application (not on a web site) which displays things like the buyer's email address [obtained via an API call] to the seller, is this a public display? It seems to be a "display of eBay Content to anyone other than the owner, originator, creator, or developer of such eBay Content".
The term can be a bit confusing. What we mean by the term "public display" is when the page will be seen by the general public. We want it to be clear that the content is coming from eBay.
If non-customers can see your pages that show content that comes from eBay then the logo needs to be displayed. If that content is only being displayed to a specific customer and it is clear in the context of the display that the content is coming from eBay then you don't need to display the logo.
For a simple example, if you have an application "SuperLister for eBay" that has a section titled "Your eBay listings" then it is obvious that the content is on eBay and you don't need to show the logo. If the pages are to create the listings that aren't yet live or work with customers inventory that is not yet on eBay you don't need to show the logo.