Show menu item only if user is logged In (not word press login)

Use the wp_nav_menu_objects filter to access the menu items

I have used this code:

add_filter('wp_nav_menu_objects', 'ad_filter_menu', 10, 2);

function ad_filter_menu($sorted_menu_objects, $args) {
    // check for the right menu to rename the menu item from
    // here we check for theme location of 'primary-menu'
    // alternatively you can check for menu name ($args->menu == 'menu_name')

    if ($args->theme_location != 'primary')  
        return $sorted_menu_objects;

    // rename the menu item that has a title of 'Log ind'
        if (is_user_logged_in()) {
            foreach ($sorted_menu_objects as $key => $menu_object) {
                // can also check for $menu_object->url for example
                // see all properties to test against:
                // print_r($menu_object); die();
                if ($menu_object->title == 'Log in') {
                    $current_user = wp_get_current_user();
                    $menu_object->title = $current_user->user_login . " - Log out";
                    $menu_object->url = wp_logout_url();;
                }
            }
        }
    return $sorted_menu_objects;
}

Found the basis for that here: Remove a menu item in menu

Obviously you need to change the line

if (is_user_logged_in()) {

to

if( isset($_SESSION['user']) ) {

and you may need to check for additional menu items to change their title and url depending on status (Logged in or out)