Drupal - How do I show the current node title in page.html.twig OR in node.html.twig?

The node is already available in the page template thanks to template_preprocess_page, so there's no need for the preprocess function. You can output the title with:

{{ node.label }}

The page title itself is a block, so if you don't need it any more just remove it from the region at /admin/structure/block. No need to copy the template to your theme or alter the original.

You can add below codes to top of node.html.twig in templates folder of your theme :

{% if not page %}
  <h2{{ title_attributes }}>
    <a href="{{ url }}" rel="bookmark">{{ label }}</a>
  </h2>
{% endif %}

{% if page %}
  <h2{{ title_attributes }}>
    {{ label }}
  </h2>
{% endif %}

The Drupal 8 code for doing what the code you shown should do is the following one.

function theme_preprocess_page(&$variables) {
  $request = \Drupal::request();
  $route_match = \Drupal::routeMatch();
  if ($route_match->getParameter('node')) {
    $page_title = \Drupal::service('title_resolver')->getTitle($request, $route_match->getRouteObject());
    $variables['my_node_title'] = $page_title;
  }
}