Fix docs copy button: run init immediately, add astro-code styles, visible by default

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

website/src/pages/docs/[...slug].astro

@@ -147,7 +147,7 @@ const sections = [
           });
         });
       }
-      document.addEventListener('DOMContentLoaded', init);
+      init();
       document.addEventListener('astro:after-swap', init);
     })();
   </script>