hook_form_alter help (almost there!)
You should be able to just copy what that linkedin code is doing in it's own linkedin_user_settings_form_submit submit function: 

$op = $form_state['values']['op'];
if ($op == $form['linkedin']['reset']['#value']) {
 drupal_set_message(t('LinkedIn preferences have been reset'));
} else {
  drupal_set_message(t('LinkedIn preferences have been saved'));
}