You may first want to look into Safari's "Installed Plugins" section within the browser. This is (usually) found under the Safari Help header in the top of a browser. I say usually, because I don't use Safari often and mine is an older version. They handle applications and plugins like most others, to a point.
I see in Safari Installed Plugins that mine would use
| application/pdf | Acrobat Portable Document Format |
Acrobat Reader & reader plugin v9.5.5 for web browsers
(in Safari 5.x, where I looked into this)
While I use FireFox, SeaMonkey (Mozilla) and others, where I can choose in preferences the helper applications to open from the browser, Safari has different methods; so if you can change this in the preferences or settings of Safari in a later version I am not certain.
You could see what the defaults are by using Safari help menu to open the Installed plugins, and maybe find out what happend to Adobe reader plugin. Could be it was 'sandboxed' in a later OS X and needs to be installed differently to be accepted as safe.
It is easier to set how a file type is handled once it is inside the computer and not held in the browser; by file type in general or specific to each single document. I do both, and seldom let an a browser do these things due to security considerations. I download and open them, only then if I feel they are safe.
Sorry to not have a specific answer for that anomaly.
Good luck & happy computing! 🙂
