Serve API and website behind Caddy path prefixes

Caddy forwards /wizard/api/* and /wizard/website/* with the prefix
stripped, so the server received /ws instead of /api/ws and the website
connected to /api/ws on the domain root.

- Server routes are now /ws and /health (was /api/ws, /api/health)
- Website derives the websocket URL from its own location (../api/ws),
  giving /wizard/api/ws in production and /api/ws in development
- Vite dev proxy strips /api like the production proxy
- nginx back to the plain SPA config since Caddy strips the prefix
- /internal/ai rejects requests that came through the reverse proxy,
  in addition to the shared secret
- Docs updated for the prefixed deployment

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

.env.example

@@ -2,7 +2,7 @@
 #   python3 -c "import secrets; print(secrets.token_urlsafe(32))"
 AI_SHARED_SECRET=change-me
 
-# Optional: comma separated list of allowed websocket origins, e.g. https://wizard.example.com
+# Optional: comma separated list of allowed websocket origins, e.g. https://game.marvinkrausser.com
 ALLOWED_ORIGINS=
 
 # Agent of the AI service: model (falls back to heuristic without checkpoint), heuristic or random