I have been able to boot it properly. Hence, I am posting below, the steps for future users. The restored partition showing up as RAW volume and Recovery partition under Windows was to be omitted. I guess, it's because Windows cannot recognize ext4 format. Macrium backup and restore had worked perfectly fine in this case.
Backup and restore of single partitions are possible. There is no need to have the full disk backed up. Nor reinstalling is necessary. Saves time and effort. If you want to save further space, or have to clone it to a smaller disk/partition, you will have to shrink the linux partition as much as possible before imaging for backup as Macrium images the whole partition bitwise (not the case for Windows).
Solution
The easiest method was to create a bootable live USB/disk of the same OS I had. In my case, I had restored the xubuntu/budgie partition image. I created a live disk of Ubuntu, and booted into it using "Try Ubuntu without installing" entry.
Gparted was able to recognize the restored partition as ext4. Now, the only issue was creating a boot menu entry for it.
For that, I followed the steps mentioned in this post How can I reinstall GRUB to the EFI partition as answered by cl-netbox.
Grub will recognize and add Windows, hence dual booting was sorted.