Thank you for posting this. The Preferences panes are indeed different. On your Mac mini, where the "Photos" item is missing, the "My Photo Stream" item is missing in the preferences. This should only happen, if you are using a new AppleID on this Mac. Are you using a different AppleID on the Mini?
Also the "Autoplay Videos" item is missing. It looks like the Photos 4 preferences panel in Mojave.
Have you tested in a different user account, if you are getting the same results, when you use a clean new user account for testing? How to test an issue in another user account on your Mac - Apple Support
Sometimes missing items in the graphical user interface are caused by corrupted preferences files. If the problem vanishes in another user account, it may help to trash the user preferences.
I just tested - when I change the language for Photos to Dutch, My Photo Steam is also missing from the iCloud Preferences on my MacBook Pro 13", ut the Photos item is there in the sidebar.