Your second system has one LVM partition (which seems to be what you want for your first system, but don't know the right steps during install). Inside that LVM partition it has swap and root.
Originally Posted by satimis
The output is completely different showing /dev/mapper. The command is the same "sudo fdisk -l"
The command is the same. The output is different because the partitioning is different. The second system has /dev/mapper pseudo disks and the first system doesn't.
It appears that fdisk -l is an obsolete tool when you have LVM. The information fdisk -l gives you for that second system is a bit misleading, especially the part
doesn't contain a valid partition table
For things that aren't supposed to contain partition tables.
If I understand correctly, GRUB does not understand LVM, so when you are using LVM you need a /boot partition outside the LVM (as that second system has) and then typically you put all your other "partitions" inside the LVM (But those aren't really partitions. They just act like partitions).
In your other thread, I said that I don't normally advise making /boot separate from root. That applies to the simple cases in which GRUB can read your root partition. If you have root inside LVM and/or one of the forms of RAID that GRUB can't read, then obviously /boot must be separate from root.