How do I get rid of the Search at the top of the search results page?
I suggest way three way 


Hide it with css ( set it to display:none)
Remove it with simple Jquery command  ( .remove)
override  search result page template


I mean duplicate page.tpl.php file in your custom theme set name to page--search.tpl.php and remove &lt;h1&gt;&lt;?php print $title; ?&gt;&lt;/h1&gt;. finally don't forget clear the cache

If this TPL doesn't work then you have to declare it in template.php using theme_hook_suggestions in the same way as answer of your linked question. See for the URL which your instance is using for giving search result.

For $base_url/search use

//replace your theme name with MYTHEME
function MYTHEME_preprocess_page(&amp;$vars) { 
  if (arg(0)=='search') {
    $vars['theme_hook_suggestions'][] = 'page__search';
  }
}


Add this in your template.php &amp; as you are suggesting TPL for page, you have to use THEME_preprocess_page($vars).