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If you had to re-enter your Download ID go to the “Extensions: Update” page and click on ‘Check for Updates’. This resets the updates already found. Why do we need to do that? You wouldn't know it, but the updates already found cache the download key which was entered when they were found.

Go to the Update Sites page and search for Akeeba Backup in the search box.

There should be only one record titled “Akeeba Backup Professional for Joomla!” with the URL https://cdn.akeeba.com/updates/pkgakeebabackuppro.xml. The Name column should match the package name for Akeeba Backup on your site (typically something starting with the text “Akeeba Backup Professional package”).

I believe that you have more than one entries. Try deleting the extra ones.

If this doesn't help, delete all update sites for Akeeba Backup. Then go to Components, Akeeba Backup for Joomla. This should create a new Update Site record. If it didn't, go back to the Update Sites page and click on Rebuild.

If this still didn't work then I think that the latter message you posted really means that Joomla is missing the cached XML manifest for the Akeeba Backup package. In this case install either the version you already have or the latest version twice in a row, without uninstalling it before or in-between.

As I've said many times in tickets and our news section, Joomla's extensions updates are very finicky. In my opinion having to look at the extension record, two different tables for update sites and updates and the XML manifest just to find if an update is available is beyond overkill, it's a clown car. There should be a single point of truth for the update source and the installed extension version, the #__extensions table. Download Keys should be added before the download, not when the update is found (something I could not do when I contributed this feature as I was not allowed to touch how the updates table work — that was two years ago, BTW, and nothing changed since). Anyway, it is what it is and we all suffer through it all.