How to use 2 twig template for for one field (or how to add a variable to a template from a template)
How to add a variable to the field template from the node template?

What you see in the node template from the field is before the field preprocess is run.

The global preprocess for fields starts like this:

function template_preprocess_field(&amp;$variables, $hook) {
  $element = $variables['element'];

  // Creating variables for the template.
  $variables['entity_type'] = $element['#entity_type'];
  $variables['field_name'] = $element['#field_name'];
  $variables['field_type'] = $element['#field_type'];
  $variables['label_display'] = $element['#label_display'];


Here can see, how the variables for the field template are built.

Example:

To add a variable in node set the variable with a '#' prefix:

function mytheme_preprocess_node(&amp;$variables) {
  if (isset($variables['content']['my_field'])) {
    $variables['content']['my_field']['#var1'] = 'a value';
  }
}


And then in the field template:

function mytheme_preprocess_field(&amp;$variables) { 
  if (isset($variables['element']['#var1'])) {
    $variables['var1'] = $variables['element']['#var1'];
  }
}


The variable is then available in the field twig:

{{ var1 }}


To solve your specific problem I would suggest you copy the field in node preprocess and then set for both copies different variables.