Threaded Comments Not Working

I am trying to enhance my theme and add support for threaded comments there. But this doesn’t seem to work. I followed the instructions at https://codex.www.ads-software.com/Migrating_Plugins_and_Themes_to_2.7/Enhanced_Comment_Display and made all the required changes:

1. I put in the loop to use wp_list_comments() with the no extra parameters in my comments.php
2. I added if ( is_singular() ) wp_enqueue_script( 'comment-reply' ); to my header.php, just above the call to wp_head()
3. I added the call to comment_id_fields(); in my comments form.
4. My comments textarea has ID = “comment”
5. My comment form is enclosed in a div with ID = “respond”

The following is the code for my comments.php file:

<div id="comments-wrap">
<?php // Do not delete these lines
	if (!empty($_SERVER['SCRIPT_FILENAME']) && 'comments.php' == basename($_SERVER['SCRIPT_FILENAME']))
		die ('Please do not load this page directly. Thanks!');

	if ( post_password_required() ) { ?>
		<p class="nocomments">This post is password protected. Enter the password to view comments.</p>
	<?php
		return;
	}
?>

<!-- You can start editing here. -->
<?php // Begin Comments & Trackbacks ?>
<?php if ( have_comments() ) : ?>
<h3 class="comments"><?php comments_number('No Comments', 'One Comments', '% Comments' );?> to “<?php the_title(); ?>”</h3>
	<div class="navigation">
		<div class="alignleft"><?php previous_comments_link() ?></div>
		<div class="alignright"><?php next_comments_link() ?></div>
	</div>

<ul class="commentlist">
	<?php
	$commentargs = array(
		"thread_comments_depth" => 5,
		"avatar_size" => 48
	);

	wp_list_comments($commentargs);
	?>
</ul>
	<div class="navigation">
		<div class="alignleft"><?php previous_comments_link() ?></div>
		<div class="alignright"><?php next_comments_link() ?></div>
</div>
<?php // End Comments ?>

 <?php else : // this is displayed if there are no comments so far ?>

	<?php if ('open' == $post->comment_status) : ?>
		<!-- If comments are open, but there are no comments. -->

	 <?php else : // comments are closed ?>
		<!-- If comments are closed. -->
		<p><?php _e('Sorry, the comment form is closed at this time.'); ?></p>

	<?php endif; ?>
<?php endif; ?>

<?php if ('open' == $post->comment_status) : ?>

<div id="respond">

<h3 class="respond"><?php comment_form_title( 'Leave a Reply', 'Leave a Reply to %s' ); ?></h3>

<?php if ( get_option('comment_registration') && !$user_ID ) : ?>
<p>You must be <a href="<?php echo get_option('siteurl'); ?>/wp-login.php?redirect_to=<?php echo urlencode(get_permalink()); ?>">logged in</a> to post a comment.</p>

<?php else : ?>

<form action="<?php echo get_option('siteurl'); ?>/wp-comments-post.php" method="post" id="commentform">

	<?php if ( $user_ID ) : ?>

<p>Logged in as <a href="<?php echo get_option('siteurl'); ?>/wp-admin/profile.php"><?php echo $user_identity; ?></a>. <a href="<?php echo wp_logout_url(get_permalink()); ?>" title="Log out of this account">Log out &raquo;</a></p>

	<?php else : ?>

	<p>
	<input type="text" name="author" id="author" class="textarea" value="<?php echo $comment_author; ?>" size="28" tabindex="1" />
	<label for="author"><?php _e('Name'); ?></label> <?php if ($req) _e('(required)'); ?>
	</p>

	<p>
	<input type="text" name="email" id="email" value="<?php echo $comment_author_email; ?>" size="28" tabindex="2" class="textarea" />
	<label for="email"><?php _e('E-mail'); ?></label> <?php if ($req) _e('(required)'); ?>
	</p>

	<p>
	<input type="text" name="url" id="url" value="<?php echo $comment_author_url; ?>" size="28" tabindex="3" class="textarea" />
	<label for="url"><?php _e('<acronym title="Uniform Resource Identifier">URI</acronym>'); ?></label>
	</p>

	<?php endif; ?>

	<p>
	<label for="comment"><?php _e('Your Comment'); ?></label>
	<br />
	<textarea name="comment" id="comment" cols="60" rows="10" tabindex="4" class="textarea"></textarea>
	</p>

<div class="cancel-comment-reply">
	<small><?php cancel_comment_reply_link(); ?></small>
</div>

	<p>
	<?php comment_id_fields(); ?>
	<input name="submit" id="submit" type="submit" tabindex="5" value="<?php _e('submit'); ?>" class="Cbutton" />
	</p>
	<?php do_action('comment_form', $post->ID); ?>
</form>
<?php endif; ?>
</div>
<?php else : // Comments are closed ?>
<p><?php _e('Sorry, the comment form is closed at this time.'); ?></p>
<?php endif; ?>
</div>

The following is the amendment in my header.php:

<?php
	if ( is_singular() ) wp_enqueue_script( 'comment-reply' );
	wp_head();
?>

And the following are the changes made in functions.php:

// Begin legacy comments support
add_filter( 'comments_template', 'legacy_comments' );
function legacy_comments( $file ) {
	if ( !function_exists('wp_list_comments') )
		$file = TEMPLATEPATH . '/legacy-comments.php';
	return $file;
}
//End legacy coments support

I have ensured that nested comments are turned on from the settings page as well.

Here is what happens:

• I can see a “Reply” link for each comment. When I click on it the comment form moves right below it. So the Javascript is working.
• When I post, however, the comment goes to the end of the list – the list is in the order of posting the comments. Also, the depth of the comments is always “depth-1”. See here: https://www.aquoid.com/sandbox/?p=188.
• I switched to the WP default theme. The results are slightly different: when I post a response to a comment, the new comment shows up where it should, but I still get “depth-1” always, so there is no nesting.

What could I be doing wrong?