Linux – Logical Volume /dev/centos/root is extended but /dev/mapper/centos-root is not

The lvextend command you executed would only extend the filesystem by 800MB (assuming default extent size), a rounding error when you're looking at TB filesystems. The -l flag means "extents" which, by default, are 4MB in size. If you wish to grow the filesystem by, say, 200GB, the command would be: lvextend -L +200G vg2/lv_backup. (Note the difference between -l and -L.)

The problem is that the initial filesystem was 1TB (almost) and the new one is 800GB, so the new logical volume should now be around 1.8TB. From here, you can see that both the drives are part of the same volume groups:

The initial filesystem was 1TB. The new disk is 800GB. You added that disk to the VG, then used 800MB of that disk to extend the LV, then grow the filesystem on the LV, leaving you with 1.0008TB in the LV. Then you grew the filesystem to fill the LV.

If you want to end up with a 1.8TB filesystem, do this:

lvextend -L +800G vg2/lv_backup

or:

lvextend -l 100%FREE vg2/lv_backup

Followed by the resize2fs command.

Every time you perform an operation with LVM, by default, the previous metadata is archived in /etc/lvm/archive. You can use vgcfgrestore to restore it, or grab the extends by hand (harder, but lvcreate(8) should cover it).

Edit:

And to make it as easy as possible, I should add that you can find the last backup before your destructive operation by looking at descriptions:

# grep description /etc/lvm/archive/vg01_*
/etc/lvm/archive/vg01_00001.vg:description = "Created before executing 'lvremove -f /dev/vg01/foo'"
/etc/lvm/archive/vg01_00002.vg:description = "Created before executing 'lvremove -f /dev/vg01/bar'"
/etc/lvm/archive/vg01_00003.vg:description = "Created before executing 'lvremove -f /dev/vg01/baz'"

Edit:

The normal allocation policy (default one) will allocate a stripe from the first free PE when there is enough room to do so. If you want to confirm where the LV was allocated, you can look in the archive files, those are perfectly readable by humans.