Drupal - How to get the valid URL of a Link field from within a Twig template?

Your PHP from your latest edit would translate into twig like this:

{{ node.field_my_link.0.url }}

This works like this, 0 returns the first item of the field item list, url gets the url object and because twig will cast this object as a string this will call the magic method toString() and will output the url as a string value.

You only need to use url, because twig looks automatically for the method getxyz() if there is no property with that name xyz.

You can access the render array element directly, in your case:

{{ node.field_my_link[0]['#url'] }}

You have to cover both cases: external and internal URLs.

First let's check if the URL is external and simply print its *.uri.

Else if it's internal we have to wire its route name and parameters through Drupal's path($name, $parameters, $options) function.

{% if node.field_link.0.url.external %}
  <a href="{{ node.field_link.uri }}">
    {{ node.field_link.title }}
  </a>
{% else %}
  <a href="{{ path(node.field_link.0.url.routeName, node.field_link.0.url.routeParameters) }}">
    {{ node.field_link.title }}
  </a>
{% endif %}