Pull Tags But Not as Links

You can use get_the_terms(). I’ve adapted the following from an example on that page:

$terms = get_the_terms( $post->ID, 'visits' );

if ( $terms && ! is_wp_error( $terms ) ) : 

    $visits_name = array();

    foreach ( $terms as $term ) {
        $visits_name[] = $term->name;
    }
    $terms_list = join( ", ", $visits_name );
    echo $terms_list;
endif;

EDIT:

Using wp_list_pluck, as suggested by Telos, is much easier:

$terms = get_the_terms( $post->ID, 'visits' );

if ( $terms && ! is_wp_error( $terms ) ) : 
  echo join( ',', wp_list_pluck( $terms, 'name' ) );
endif;