I can’t write in my DB using $wpdb->insert

I would say the problem is first array. Change it to

array(
    'name' => $_POST['name']
),

You can also use wpdb->show_error() or wpdb->print_error()

To Samuel Elh: wpdb->insert() can escape value itself

Hata!: SQLSTATE[HY000] [1045] Access denied for user 'divattrend_liink'@'localhost' (using password: YES)