Snapper automatically creates everything it needs to hold the snapshots. You shouldn’t need those @snapshots subvolumes and you don’t need to create anything special for it.
lsblk shows block devices. btrfs subvolumes aren’t block devices so they shouldn’t show there.
You can use btrfs subvolume list / to see subvolumes/snapshots. What lsblk will show you is all the mountpoints tied to a block device so if you mount the subvolumes lsblk will list the mountpoints.
Do you have a subvolume mounted on /var? That will make it harder to recover if you ever need to rollback a snapshot. You probably don’t want that on Arch-based system.