Very easy, for beginners like me:

To identify the start page, use this php code in your template:

if(return_page_slug() == "index" ) {
  // do something or not
}

PS: I came across this easy solution by checking the default theme "Innovation" where the breadcrumbs are hidden by CSS from the index page (see also header.inc.php there). I thought it is better not to show them at all...