There's no [Playlist] field as you've discovered. I believe that MC simply views the collection of files or files referenced by lists in the sync list as one singular list. So there is no name available.
Since tracks may be in more than 1 playlist, and syncing can select from one or more playlists, it isn't clear what should be done in the case where you sync to a device from say five playlists and a track is in several of them. Should the track be duplicated to the single device in each of the relevant [Playlist] folders?
I'm guessing that [Genre] won't work for you and that your description used a simple example. Maybe there is some other field that is unique and useful.
You could simulate your own [Playlist] property in the simple case (but this method will have serious shortcomings). Create the user field, something like [Playlist Name], select the tracks in that playlist before syncing, and assign a value to the tracks. Do this either in-column (add the [Playlist Name] column for easy identification), or in the Tag Action Window. This clearly won't work for automatic syncing when a sync list is comprised of more than one playlist and any track is one more than one list. Also, it requires you to pre-set the [Playlist Name] field before syncing.