Display taxonomy term in Drupal Twig template translated

node--talk--teaser.html.twig

{#
/**
 * @file
 * Theme override to display a Talk node in Teaser view mode.
 *
 * The 'talk_category' variable was created in preprocess.
 */
#}

{# Other variables are excluded from this example to keep it simple. #}
{% include "@molecules/card/01-card.twig" with {
  card_title: tt_category,
} %}

page--talk.html.twig

{#
  It is a simple solution if there are not to much languages to support.
  The example uses a component based aproach. Look for Emulsify theme for more info.
#}

{# Get the translated taxonomy term. #}
{% set pill_text_nl = node.field_talk_category.entity.translation('nl').name.value %}
{% set pill_text_fr = node.field_talk_category.entity.translation('fr').name.value %}

{# The next definition uses the Global variables module. #}
{% set current_lang = global_variables.current_langcode %}

{# Set the correct translation depending on the actually used language. #}
{% if current_lang == 'nl' %}
  {% if pill_text_nl %}
    {% set pill_text = pill_text_nl %}
  {% endif %}
{% elseif current_lang == 'fr' %}
  {% if pill_text_fr %}
    {% set pill_text = pill_text_fr %}
  {% endif %}
{% endif %}

{# Other variables are excluded from this example to keep it simple. #}
{% include "@pages/_page.twig" with {
  hero_pill_text: pill_text,
} %}

THEMENAME.theme

<?php

/**
 * Implemets hook_preprocess_node().
 *
 * Create variable from translated taxonomy term name.
 * Code by @tikaszvince.
 */
function THEMENAME_preprocess_node(&$variables) {
  /** @var Drupal\node\Entity\Node $node */
  $node = &$variables['elements']['#node'];
  if (
    $node->hasField('field_talk_category')
    && !$node->get('field_talk_category')->isEmpty()
  ) {
    /** @var \Drupal\taxonomy\TermInterface $category */
    $category = $node->field_talk_category->entity;
    $current_langcode = $node->language()->getId();
    if ($category->hasTranslation($current_langcode)) {
      $category = $category->getTranslation($current_langcode);
    }

    $variables['#cache'] += ['tags' => []];
    $variables['#cache']['tags'] = array_merge($variables['#cache']['tags'], $category->getCacheTags());
    $variables['talk_category'] = $category->label();
  }
}

Good to know @bmsimo! Also thanks for the update! :)