changing behaviour of get_search_form

you don’t need to add another function to get the search form. If you want to change the look of the search form or add another items you’ll have to add the searchform.php to your theme.

It could look like this:

    <form role="search" method="get" class="search-form" action="<?php echo home_url( "https://wordpress.stackexchange.com/" ); ?>">
        <input type="search" class="search-field" placeholder="Suchen …" value="<?php echo get_search_query() ?>" name="s" title="Suche nach:" />
        <input type="submit" class="search-submit" value="Suchen" />
    </form>

Everytime you call the get_search_form() function it will return the content of the searchform.php file. There you can add as many form fields as you wish.