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.