No … update-grub is one thing.
It is an executable script in /usr/bin that looks like this
#! /bin/sh
set -e
grub-mkconfig -o /boot/grub/grub.cfg "$@"
# workaround for https://git.savannah.gnu.org/cgit/grub.git/commit/?id=3cf2e848bc03c44d30bb87e583d12efe7e7ccf75
# if grub is not updated in MBR/EFI dashes still won't work, hence we remove them
sed -i -e '/cryptomount -u/ {s/-//g;s/ u/ -u/g}' /boot/grub/grub.cfg
And what it does is pretty much apply settings. Such as after you made changes to boot options.
But when ‘people’ refer to ‘updating your grub installation’ … ie - the software version of grub that runs whenever you turn your computer on … reformatting it with version 2.5 over version 1.7 … that is not update-grub … it is a command that is system-dependent and would look something like
sudo grub-install --target=x86_64-efi --efi-directory=/boot/efi --bootloader-id=Manjaro --recheck
(but phil has recently pushed a new script that is attempting to automate that called install-grub)
The article is simply wrong… Or maybe just really misleading.