[codex] Update actions cache to v6

[amyu]

Summary

• Update @actions/cache from ^5.0.3 to ^6.1.0.
• Replace the custom patch-package patch with a v6.1.0-compatible patch that keeps CacheEntry metadata available.
• Regenerate the bundled dist/setup and dist/cleanup outputs.
• Explicitly include Node types in tsconfig.json for the newer action dependency type surface.

Impact

This keeps the action on the current @actions/cache major while preserving the existing cache summary behavior that depends on cache key and size metadata.

Validation

• ASDF_NODEJS_VERSION=24.15.0 npm ci
• ASDF_NODEJS_VERSION=24.15.0 npm run all
• ASDF_NODEJS_VERSION=24.15.0 npm audit --audit-level=high
• git diff --check