Why do my iPhone playlists not show up under "My Device" on Mac computer? When I manually connect my iPhone to my Mac Pro, click on My Device, it does not show my  iPhone playlists. My goal is to move specific playlists over to computer so I can play them on both.
You shouldn't click on My Device. Just plug in your iPhone, make sure you are syncing Music, then apply all settings, then click Sync. 

After sync, go back to your Music, at the bottom you will see all your playlists. I just made a playlist called Test for this answer, see below:



Update: New iTunes have iCloud Music Library, check if it works when you enable it: