What’s the equivalent of front-page.php for a custom static page?

The Template Hierarchy only covers what’s listed here. So there is no filename you can give a template for it to be automatically used by default.

However, it is possible to add your own logic to the template hieararchy with the page_template_hierarchy hook. So using this hook, you could check if the current page being viewed is the page that has been set as your custom static page, and if so, add your own template name to the top of the list.

So, in the accepted answer at your link, the static page is saved as page_for_projects, which means the code would look like this:

function wpse_339118_projects_template_hierarchy( $templates ) {
    $page_id           = get_queried_object_id();
    $page_for_projects = get_option( 'page_for_projects' );

    // If the current page is the Projects page.
    if ( $page_id === $page_for_projects ) {
        // Use projects.php as the template, if it exists.
        array_unshift( $templates, 'projects.php' );
    }

    return $templates; 
}
add_filter( 'page_template_hierarchy', 'wpse_339118_projects_template_hierarchy' );

With that code, if you name your template projects.php, it will be loaded automatically when viewing the page that’s saved as your custom static page.