Adding a third custom taxonomies

For every taxonomy you have to choose a unique name, which is used in the database. so so your new e.g. players_region.

<?php
add_action( 'init', 'build_taxonomies' );

function build_taxonomies() {
    register_taxonomy(
        'players_team', // Unique name.
        'player',       // Post type.
        array(
            'hierarchical' => true,
            'label' => 'Players Team',
            'query_var' => true,
            'rewrite' => true,
        )
    );
    register_taxonomy(
        'players_region',
        'player',
        array(
            'hierarchical' => true,
            'label' => 'Players Region',
            'query_var' => true,
            'rewrite' => true,
        )
    );
}