Drupal - Check if it's the front page in the template

The variable you're looking for is called is_front:

{% if is_front %}

The available variables for a template are documented at the top of the .html.twig file, however there is also a set of default variables available to all templates (which are not documented in every single template). You can find them in _template_preprocess_default_variables().

Note that if you want to provide $variables['is_front'] to templates that doesn't, you can add this into the concerned preprocess function.

function themename_preprocess_menu(&$variables) {
  try {
    $variables['is_front'] = \Drupal::service('path.matcher')->isFrontPage();
  }
  catch (Exception $e) {
    $variables['is_front'] = FALSE;
  }
}

It is done the same way in template_preprocess_page for page.html.twig.

Same answer here

Assuming that the template you are using is page.html.twig, then the variable you are looking for is is_front; front_page contains the URL of the front page, initialized in template_preprocess_page() with the following code.

$variables['front_page'] = \Drupal::url('<front>');

For other template files, Drupal doesn't provide any variables telling you if the user is visiting the front page. Using code similar to the one used from template_preprocess_page(), you can set that variable in the preprocess function for the template you are using.

function mymodule_preprocess_HOOK(&$variables) {
  // An exception might be thrown.
  try {
    $variables['is_front'] = \Drupal::service('path.matcher')->isFrontPage();
  }
  catch (Exception $e) {
    // If the database is not yet available, set the default value.
    $variables['is_front'] = FALSE;
  }  
}

Replace HOOK with the template type (e.g. field for the field.hmtl.twig template).