Written like this, your theme will load the menu 'header_menu' if exists (and there is probably no reason why users should name their menu this way) or always loads the first created menu. That is probably what happens to you.
There is no other way than modifying your theme and, as always, the best is to take default themes as example.
In your file functions.php, add this line (for example just after the first line <?php):
register_nav_menu( 'primary', 'Header menu' );
And modify the line of code above in:
<?php wp_nav_menu('theme_location=primary&container=false&menu_id=menu'); ?>