Define a Custom Template Part As A Widget Area

You should simply need an appropriate dynamic_sidebar() call in your template.

https://developer.wordpress.org/reference/functions/dynamic_sidebar/

You pass the ID of the sidebar you registered it with using register_sidebar.

https://developer.wordpress.org/reference/functions/register_sidebar/

You may wish to use some logic to see if the sidebar is active first.

$has_sidebar_one = is_active_sidebar( 'first-sidebar-id' );

And later in the appropriate area you want to render the sidebar..

<?php if ( $has_sidebar_1 ) : ?>
<div class="my_sidebar_element">
    <?php dynamic_sidebar( 'first-sidebar-id' ); ?>
</div>
<?php endif; ?>

Hope that helps.