Backup Pi onto same SD card, different partition

As far as I understand you want to have a disaster recovery. It is possible to do it with the steps you described and with some scripts. The main problem is that you cannot simply clone the running operating system. The copy takes some time and within this time the OS will change several system files that may result in an inconsistent copy. It is possible that the copy does not work. You have to take additional effort to avoid changes during copy. You can use rsync as described at Can a Raspberry Pi be used to create a backup of itself?.

But I prefer to use the Logical Volume Manager (LVM). With it you can just take snapshots at any time with a one liner sudo lvcreate --snapshot --name current_setup --size 4g rpi_vg/root. Now you can reconfigure your running system until it is broken and does not boot again. You can boot into the snapshot and repair the main logical volume (lv) by mounting it. Or you can just revert the main lv to the current_setup with a one liner sudo lvconvert --merge rpi-vg/current_backup. I always use LVM for my development. There are many other nice things you can do with it, e.g. easy taking backups. Have a look at Easy backups and snapshots of a running system with LVM.

You cannot install Raspberry Pi OS into a partition - it needs 2 partitions.

You can copy the root partition into a separate partition (although this requires you to modify the partition layout - which can't be done on a live system) and easily switch between the two (provided they share a kernel).

I have done this, but doubt it makes much sense.

NOTE if you want to do this you need to ensure that the root partition in /etc/fstab is changed (probably -03) and select the partition to boot in cmdline.txt

PARTUUID=b5d2de0f-02  /               ext4    defaults,noatime  0       1