How to duplicate a page template but make minor changes to the header?

Your best bet is to rename the old header.php to header.bak.php and your new copy to header.php. header.php (there’s only one such file, always) is called from the other theme templates using get_header(). What you could/eventually should do, if you don’t want to directly edit the existing theme, is set up a child theme, copy over header.php to that child theme’s directory, and edit it there. If the theme was custom-made and isn’t going to be updated anyway, you might however skip that part. Just make sure there’s a backup.