How do I tell Drupal 8 to use my custom module's theme template when rendering an image that is linked?
Answering my own question. The solution was to use: hook_theme_registry_alter() 

/**
 * Implements hook_theme_registry_alter
 */
function MY_MODULE_theme_registry_alter(&amp;$theme_registry) {
  $theme_registry['image_formatter']['path'] = drupal_get_path('module', 'MY_MODULE') . '/templates/field';
}

In my module I have the templates/field directory with the template image-formatter.html.twig, so no other changes were necessary.