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These are two completely separate things. I think you are confusing two entirely different features.

Backup quick icon

First of all, you need to publish the plugin Quick Icon - Akeeba Backup Notification.

This plugin tells Joomla to set up the backup icon in your dashboard's Notifications section. This is where you get the icons for Joomla and extensions updates, download keys in need to be entered, overrides to check, and urgent privacy requests. All of these icons are plugins. Most of these are shipped with and enabled by Joomla. The backup icon is, of course, a plugin we ship with our software.

In itself, the Notifications section is an administrator module published in the icon position, and has the Group set to update_quickicon. If you have unpublished, deleted, or changed this administrator module then you will see no notification icons, including the quick backup icon. I believe that if you don't see the backup icon you have either unpublished the plugin OR unpublished, deleted, or changed the administrator module.

The point of this icon is to check how long ago you've taken a successful backup and notify you if a backup is necessary.

Backup on update

You need to publish the plugin System - Backup on update.

This will only display a notice about the automatic backup on update at the top of the page when you go to System, Update, Joomla. It does not display an icon in the Joomla Home Dashboard page.

The point of this plugin is to let you run a backup automatically when updating your site through the Joomla Update core component, right after downloading the update package and before it's actually installed on your site.