Hi @chrisj5 ,
The difference between the embedded PDFs in these stories is where the PDFs are hosted. In Story 1, nothing appears when the "2023 Washington State Enhanced Hazard Mitigation Plan" card is clicked because that PDF is not hosted in ArcGIS Online. When you host your PDFs in ArcGIS Online (like the "Impervious Surface Elementary Schools Grounds in Pierce County" PDF in Story 2), the only option to link to that PDF is through a download. To prove this, if you go to your ArcGIS item details page for that PDF, you will see no option to view that PDF, only to download.
Because of this, this is more of an ArcGIS Online behavior to address than an ArcGIS StoryMaps behavior. It looks like this was actually a recent, intentional change by ArcGIS Online for more secure practices. See the full description on ArcGIS Online's Esri Community here.
To at least workaround this automatic download in your story, host the PDF outside of ArcGIS Online and use that external link in your embed (like you do in Story 1).
Let me know if this works for you!
--Abby