How to include custom code on a page that’s set to act as homepage?

Since you are saying it’s a “Page”, I believe you should try is_page instead of is_home:

<?php
    if ( is_page( 'about' ) ) {    
         // Code to be shown on Page whose slug is "about"
    }
?>

OR for the way your code is, this should do:

<?php if ( is_page( 'about' ) ) { ?>
    <!-- Code to be shown on Page whose slug is "about" -->
<?php } ?>