docs: refresh timestamps across Betti-RDL docs

[cto-new]

Summary

Refresh Last Updated timestamps across Betti-RDL documentation and README to reflect the latest changes and date of last push.

Details

• Update top and bottom dates in COMPREHENSIVE_RSE_REVIEW.md to current date (2025-12-18).
• Update API_REFERENCE.md, GETTING_STARTED.md, ARCHITECTURE.md, TROUBLESHOOTING.md, CONTRIBUTING.md Last Updated sections to current date.
• Update PHASE3_COMPLETION_REPORT.md completion date to current date.
• Update README.md to include Last Updated badge with the current date.
• Apply consistent metadata headers to documentation files as per the standard format.
• Verify no documentation contains future dates and that all dates reflect the latest changes.
• Note: No breaking changes; this is a documentation-only update.