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When restoring, there is a tick box in the "Site Setup" page of Akeeba Backup Installer which allows you to use the tmp and log directories under your new site's root. When moving your site around, it's a good idea to tick that box. It will save you from the trouble of having to do that manually :)

Now, regarding your problem, it's a little different than what you think. Akeeba Backup will backup all files and folders under your site's root. This means that when backing up the site in your server's root it will backup all of its folders, including the subdirectory where the old site exists. You have to use the Files and Directories Exclusion button in Akeeba Backup's main page to exclude the old site's subdirectory from the backup.

The reason Akeeba Backup works that way is very simple, actually. There is no way to tell within any degree of precision if a subdirectory whose name doesn't belong to the core Joomla! installation is used by your site or contains another site / third party application / is otherwise not used. Therefore, the default action is to backup everything which is what 80% of users want to do anyway. For the rest of our users, the exclusion features can be used to tell Akeeba Backup what it should not back up.