It took me hours to understand what I'm embarrassing, so I just wanted to share this with anyone looking for this in the future, because I could not find a single answer to this question at any time. You just need to set the "backup-dir" parameter in the drushrc.php file as follows:
$options['backup-dir'] = '/your/new/backup-dir';
Pay attention to the trailing slash.
I hope someone finds this helpful.
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