Ok so I've got a SharePoint 2013 environment on Server 2012 R2 and Adobe Acrobat 2017 standard edition on Windows 10
Working with Microsoft I've tweaked SharePoint so that when you click on a link to a pdf within SharePoint then it opens within Adobe Acrobat outside of the browser without issue.
When I select the same file and click edit though I get a pop-up message saying "We're sorry, we couldn't find a program to open this document".
According to the Microsoft tech the outstanding edit issue is because in the Manage-Add On section of IE / Edge the add on "Adobe Acrobat SharePoint OpenDocuments Component" is installed, but disabled and the option to enable is greyed out.
He indicates if I can get this add-on to work my error should go away. The difficult part is this is deployed enterprise why so I have to come up with a deployable solution.
Which web browser are you using? if it's Edge then you should know that Microsoft Edge doesn't allow plug-ins. If it's IE then it may be likely related to the method used to Open outside of the web browser being used.
DOCICON.xml is usually the item that provides the context menu items on a SharePoint web site. Provided this configuration that you setup is only on the SharePoint site and not also in part also via IE, then you can test that hypothesis. Go to a different SharePoint site that allows Opening the PDF in the web browser and doesn't force the opening outside the web browser. Then test the edit capability.