Why do my iPhone playlists not show up under "My Device" on Mac computer?
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: