I believe that the reference to 13565 is the total number of items (songs and other media, such as Podcasts) that iTunes is managing on your iPod, while the 12580 figure is the number of items that iTunes is dealing with at this specific Sync. Compare your view with mine showing what iTunes says as my iPod was being synchronised:

Notice that iTunes reports that it is copying 18 items, but the Sync Music figure is 4179 songs. So the number of songs iTunes is managing on my iPod is 4,179, but iTunes is only working on 18 songs at this time. That may be updating a Playlist, or synchronising other information such as changed song rating, last played time, etc. Don't forget, a Sync means exactly that; synchronising, or making the same. That includes information about the song itself, not simply adding it to a device.
When I check the iPod's Settings/About>Songs, the figure is 4,198 (complete with comma this time). The discrepancy is due to other items, such as Podcasts etc. I would take those figures with a pinch of salt, they never make sense to me and I've tried to understand them over many years.
If a song is listed on your iPod, but with a cloud with arrow in it, that means that it is not downloaded to your iPod, but it is a song you have purchased from the iTunes Store. The U2 album Songs of Innocence is listed on my iPod like this. That's because it's in my iTunes Store account as a purchase (even though I didn't purchase it - it was an unwanted gift from Apple and U2), but I have chosen to not download it to my iPod. Tap the download arrow to download the song. Better still, make sure that you download those purchases to your iTunes Library and then let iTunes add them to the iPod from there.