refactor: extract executeConversationTurn() helper and fix transient timing

[saraichinwag]

Summary

Extracts a shared executeConversationTurn() private method from the three chat endpoints (handle_chat, handle_continue, handle_ping), eliminating ~200 lines of duplicated AI loop + error handling + AgentContext management code.

Transient Timing Fix

Moves the X-Request-ID transient cache set to before the AI loop executes (right after session creation), preventing duplicate sessions when retries arrive during processing.

Changes

• Added: executeConversationTurn() - encapsulates AgentContext set/clear, ToolManager, AIConversationLoop execution, error handling, and session error updates
• Refactored: handle_chat(), handle_continue(), handle_ping() to use the shared helper
• Fixed: Transient set timing in handle_chat() - now set immediately after session resolution, before AI loop

What's Unchanged

• All public API contracts (request/response shapes)
• Route registration and permission callbacks
• verify_ping_token, list_sessions, delete_session, get_session
• Pending session deduplication logic in handle_chat

Fixes #211
Fixes #5