[Fix] Refactor llm_search_context to use lua_cache module

[Fix] Refactor llm_search_context to use lua_cache module

- Replace manual Redis operations with lua_cache API for better consistency
- Use messagepack serialization and automatic key hashing
- Fix Leta Mullvad API URL to /search/__data.json endpoint
- Add search_engine parameter support
- Remove redundant 'or DEFAULTS.xxx' patterns (opts already has defaults merged)
- Add proper debug_module propagation throughout call chain
- Improve JSON parsing to handle Leta Mullvad's nested pointer structure