Although the title says move the steps can also be used to simply copy the library. You don't have to delete the original.
When you say cloud connects the two computers this would only be relevant if you had iTunes Match or Apple Music (the subscription service), in which case your playlists should have come over already.
I can't tell exactly what steps you've taken so far but I suspect you've imported the music into a new empty library rather than copying the entire library database over. Are you moving between iTunes and Music, or is the older computer running Catalina or later?
In either case, unless your library has something complicated going on, it should be a matter of copying the entire Music folder from your profile folder on the old computer into your profile folder on the new computer, either by way of an intermediate copy on an external drive, or over your network.
Once that has been done press and hold down option as you launch Music and use the Choose Library... option. If your old computer is running iTunes you need to open ~/Music/iTunes/iTunes Library(.itl), then choose a location for the converted library (which should be ~/Music/Music). Otherwise you need to open ~/Music/Music/Music Library(.musiclibrary).
tt2