wp_insert_post not updating custom taxonomy selected if logged in as a subscriber

It’s because within wp_insert_post current user capabilities are checked before adding the terms:

if ( current_user_can($taxonomy_obj->cap->assign_terms) )
    wp_set_post_terms( $post_ID, $tags, $taxonomy );

to get around this, use wp_set_object_terms instead after wp_insert_post to add the terms:

$new_post = array(
    'post_title' => $postTitle,
    'post_content' => $post,
    'post_status' => 'publish',
    'post_date' => date('Y-m-d H:i:s'),
    'post_author' => $user_ID,
    'post_type' => 'publications'
);
$new_id = wp_insert_post( $new_post );
wp_set_object_terms( $new_id, $term_id, 'publicationstype' );