Use header option on wp-login page with a second logo

For anything other than simple CSS tweaks your best bet is to use wp_login_form() in a custom Page Template and filter login_url to return that instead of wp-login.php:

add_filter( 'login_url', 'wpse276598_login_page', 10, 3 );

function wpse276598_login_page( $login_url, $redirect, $force_reauth ) {
    return home_url( '/my-login-page/?redirect_to=' . $redirect );
}

If you wanted to redirect all successfully logged-in users to the homepage:

function wpse276598_login_page( $login_url, $redirect, $force_reauth ) {
    $redirect = get_home_url();
    return home_url( '/my-login-page/?redirect_to=' . $redirect );
}