You have to supply the_field function with the proper ID value when getting fields associated with a taxonomy term. See the end of “usage” on this page http://www.advancedcustomfields.com/resources/functions/the_field/.
If this function is called on only tax/term templates you can use the following to get the current term id.
$term_id = get_queried_object()->term_id
The you combine that with the taxonomy name to get the field value
$post_id = ‘taxonomy-name_’.$term_id.
