WordPress Login/Logout Single Menu Only

Change the filter hook to wp_nav_menu_{menu-name}_items, so it will only run on a specific menu. For example, if the menu name is “Secondary”, use the hook wp_nav_menu_secondary_items, or if the menu name is “Top Navigation”, use wp_nav_menu_top-navigation_items (replace spaces in the menu name with dashes).

Your updated code, assuming the menu name is “Secondary” (the menu name, not the theme location), would be:

add_filter('wp_nav_menu_secondary_items', 'add_login_logout_link', 10, 2);
function add_login_logout_link($items, $args) {    
    ob_start();
    wp_loginout('index.php');
    $loginoutlink = ob_get_contents();
    ob_end_clean();         
    $items .= '<li>'. $loginoutlink .'</li>';    
    return $items;
}