Q: Itunes not detecting phone playlists

iPhone and the Mac's iTunes library can have different playlists.  It depends on several factors, such as iCloud Music Library, automatic syncing vs manual loading, etc.

A playlist on the iPhone does not necessarily have to appear in iTunes on your PC.  It depends on how things are currently set up.  If you subscribe to Apple Music, you have an iCloud Music Library.

Is iCloud Music Library turned ON on the iPhone?  If it is, are you signed IN in iTunes (on the PC)?  Are you using the same Apple ID (on the PC) that you use on the iPhone?  And is iCloud Music Library turned ON in iTunes (on the PC)? 

Do your PC and iPhone currently have an active Internet connection?

If you don't subscribe to Apple Music (or iTunes Match), you don't have an iCloud Music Library, so iCloud is not involved.  And your Apple ID is not involved.  It is simply a matter of syncing between your computer (running iTunes) and your iPhone.

I assume you connect iPhone to computer to since with your iTunes library...  How is iTunes set up to sync or load iPhone?  Do you use the Manually manage music and videos setting on the iPhone's Summary settings screen in iTunes?  Or do you use the Sync Music setting on the iPhone's Music settings screen in iTunes?

If you Manually manage, iTunes is not keeping songs (and playlists) on the iPod "in sync" with your iTunes library.  Your computer and iPhone have separate libraries.

Would this have anything to do with why the playlists listed under On my Device  in Itunes do not include the one I created through the Music app on the phone?

I'm not sure about that one...  If you want to manually create a playlist on the iPhone (that is seen in iTunes), you do it in iTunes.  Select iPhone in the iTunes sidebar (under Devices), and from the menu bar, File -> New -> Playlist.  The new playlist appears on the iPod (where you can name it), and you manually add (drag and drop) songs on that playlist.

You may need to recreate them manually.  iTunes manual loading is done in iTunes, so I don't know how playlists created on the iPhone are treated, if you don't have an lCloud Music Library (accessed by both iPhone and iTunes library on computer).

To make it easier, you could have you iPhone disconnected, showing the playlist on its screen.  Then, re-create each playlist in your iTunes library and manually populate them there (not on the iPhone).

If those playlists have all the songs you want on iPhone, you can then use automatic syncing (instead of the Manually manage setting).  That would make it more convenient going forward, after initial work of setting up the playlists in your iTunes library... 

Go to iPhone's Music settings screen in iTunes and set it to Sync Music.  This turns ON automatic syncing.  Choose option to sync Selected playlists, artists, albums, and genres.  Find those playlists under Playlists, and select (checkmark) them.  Click Apply to sync.

NOTE:  Any songs on iPhone that are not in your iTunes library will be lost.  After syncing, the only songs on iPhone are the songs on the selected playlists.

Going forward, you can update those playlists in your iTunes library (add/remove songs).  The iPhone does not need to be connected.  The next time you connect your iPhone (or click the Sync if already connected), iTunes automatically updates iPhone with the same changes to those playlists.