Hopefully gnome disks is misreporting. You should be able to open your linux partition in Caja from the live DVD to confirm that something is there.
Edit: Noticed in your screenshot sda4 is a extended partition, not a ext4 partition. This is why your getting the wrong fs type error. Run sudo fdisk -l in the terminal to find your Linux partition.
The procedure I use to repair Grub from a live DVD differs slightly from yours.
For example:
sudo mount /dev/sda1 /mnt
sudo grub-install --boot-directory=/mnt/boot /dev/sda
https://ubuntu-mate.community/t/repair-grub-2-from-a-live-dvd/13125