Hi Sven W,
The difference is because when open the document from the library, it use the “OpenDocument Class” add on, which prompt open in Read-only or Edit mode, you can use F12 to check the properties of the opening document in these two different ways.
When open the document in Search, open it by checking “Open”, people with edit permission would be able to edit the document, and save it back to SharePoint library.
Thanks,
Qiao
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