How to upgrade Drupal core manually or via composer?
Upgrading Drupal Core manually is not less than nightmare, therefore is better that you install Drupal via Composer.
When upgrading Drupal Core, likewise I am upgrading Drupal core 9.0.7 to Drupal core 9.1.2, to do so I simply ran composer update that will eventually upgrade all the packages along with drupal/core.
composer update
In case you encounter the permission issue while running composer update, for example in image blow.
To resolve this problem make sure you have proper permissions on sites/default by running following command.
chmod u+w sites/default
Once permissions are set correct, Drupal Core will be upgraded successfully.