docs: fix stale file paths from package split, add missing index entries (#937)
mood-search.md and genre-classification.md still referenced pre-split
paths (_internal/background_tasks.py, _internal/*_client.py) that moved
to _internal/tasks/ and _internal/clients/ in PR #886.
Also adds playlist-recommendations.md, liked-tracks.md, and
atproto-identity.md to the docs/README.md index.
Co-authored-by: Claude Opus 4.6 <noreply@anthropic.com>
authored by
zzstoatzz.io