Can't edit iTunes playlist

Just to make sure, are you looking at the Playlists in your Library, under the heading Music Playlists, rather than under Devices (which is "on the phone, but through the PC")? See the screenshot:

You should be looking under the Music Playlists heading. These are the playlists lists that you should edit. Then you Sync those changes (edits) to the phone by using iTunes' Sync. If you cannot edit them, check to ensure that the songs in the Playlist will play. If they don't, that's the issue.

turingtest2 has explained the difference between your two devices. In other words, the iPod appears to be set to Manually Manage, while your phone is set to Sync.

Therefore, you cannot edit the playlists directly on your phone and your screenshot shows that you are currently viewing the phone:

Indstillinger/resume

På min enhed

(No, I don't speak Danish. Google Translate is my friend)

• Tap the back arrow button at the top of your iTunes window to return to your library:

• you will now see a screen like the one I posted in my first post
• look for the heading Musik afspilningslister (if that is not Danish for Music Playlists, my friendship with Google Translate may falter)
• edit your playlist under the heading Musik afspilningslister and then use Sync to cause those changes to take effect on your phone
• to add that change to your iPod, which you appear to have set to Manually Manage, you will need to delete the Playlist on the iPod wand then drag the new one onto it in its place

I assume that you mean you can see the playlists on your phone.

However, what you need to do is move (or copy) your music folder (that contains all your music and the iTunes control files) from your old computer to the new one and then use the new library (with your old content) to manage your phone.

iTunes will let you directly edit playlists on a manually managed iPod.

tt2