How to alter Summary

This issue is clearly about altering the Paragraph Summary - aka, the closed "preview" of field values when editing some entity with paragraphs.
#2 is the UI description when checking the paragraph list, not related.
And altering this still seems like a big question mark - there seems to not be a straightforward way of doing this as you would, for example, with preprocessing variables.
There seems to be a preprocess summary method, which you can technically use to change the summary... However, it only contains the text elements and not the entity in question, so you cannot easily append it without fetching the entity from route, then loading/digging however deep you need with tons of if's.

/**
 * Prepares variables for.
 *
 * Default template: paragraphs-summary.html.twig
 *
 * @param array $variables
 *   An associative array containing:
 *   - buttons: An array of buttons to display in the modal form.
 */
function template_preprocess_paragraphs_summary(&$variables) {
  $variables['content'] = $variables['element']['#summary']['content'];
  $variables['behaviors'] = $variables['element']['#summary']['behaviors'];
  $variables['expanded'] = !empty($variables['element']['#expanded']);
}

Yes, I'm wondering why the paragraphs entity is not attached to the variables that reach the preprocess and twig template.

I propose that in ParagraphsSummaryFormatter->viewElements(), the entity, or the entity ID/Revision is attached to $elements['#attached'].

Speculative to try it out...

/**
 * Implements TEMPLATE_preprocess_paragraphs_summary
 */
function mymodule_preprocess_paragraphs_summary(&$variables) {
   var_dump($variables['element']['#paragraph']);
}

Using the module paragraph_view_mode.

/**
 * Implements TEMPLATE_preprocess_paragraphs_summary
 */
function MYMODULE_preprocess_paragraphs_summary(&$variables) {
  // Requires https://www.drupal.org/project/paragraphs/issues/3157819#comment-15149289
  $paragraph = $variables['element']['#paragraph'];
  $view_modes = \Drupal::service('entity_display.repository')->getViewModeOptionsByBundle('paragraph', $paragraph->getType());
  $view = $paragraph->get('paragraph_view_mode')->value;
  $variables['content'] = array_merge(['as ' . $view_modes[$view]], $variables['content']);
}

Patch for 8.x-1.16 with the approach suggested by @sime in #9 is provided.

The last submitted patch, 11: add_the_paragraph_to_summary_preprocess_variables-3157819-11.patch, failed testing. View results →
- codesniffer_fixes.patch Interdiff of automated coding standards fixes only.

Summary renderization was not covered in all cases in the patch #11. New patch attached.

The last submitted patch, 13: add_the_paragraph_to_summary_preprocess_variables-3157819-12.patch, failed testing. View results →
- codesniffer_fixes.patch Interdiff of automated coding standards fixes only.

Hi,

Thanks for the patch comment 13 (and also the work done previously in the issue too!).

Impossible to customize properly paragraphs summary without that.