Getting node view mode in field preprocess
In a field template the view mode is stored here:

$variables['element']['#view_mode']


So you could check in preprocess, if the node is not displayed as teaser: 

function MY_THEME_preprocess_field(&amp;$variables) {
  if ($variables['element']['#view_mode'] != 'teaser')  {
    // node is not displayed on the front page
  }
}


The same condition in a field twig:

field.html.twig

{% if element['#view_mode'] != 'teaser' %}
  node is not displayed on the front page
{% endif %}