bug: async chat with save_to_session completes but never creates a chat session

[vybe]

Summary

Async-mode chat (dynamic-thinking / fire-and-forget submission) with save_to_session=true completes the execution successfully but never creates a chat session — the conversation is silently lost instead of being persisted.

Context

Caught by two consecutive full-suite runs (2026-07-03, 2026-07-04):

• tests/test_dynamic_thinking_status.py::TestAsyncModeSessionPersistence::test_async_mode_with_save_to_session (tests/test_dynamic_thinking_status.py:455) — 0 sessions found before AND after the async turn
• ::test_async_mode_session_contains_messages (tests/test_dynamic_thinking_status.py:514) — same

Environment cascade is ruled out: in the 2026-07-04 run the sibling async tests in the same module passed (test_async_mode_returns_accepted, test_async_mode_execution_id_is_pollable, test_async_mode_completes_eventually, test_sse_stream_available_during_execution) — so the async execution path itself worked while session persistence did not.

Acceptance Criteria

• An async-mode chat with save_to_session=true creates a chat_sessions row and persists both user and assistant messages once the execution completes
• Both TestAsyncModeSessionPersistence tests pass in the integration suite
• Root cause documented (was persistence skipped on the async completion path, or does session creation race the 202 return?)

Technical Notes

• save_to_session is handled in src/backend/routers/chat.py (model in src/backend/models.py)
• Suspect the async completion path finalizes the execution without running the session-persistence step the sync path runs — check where the sync path writes chat_sessions/chat_messages vs what the async terminal applier does

[AndriiPasternak31]

/claim

[github-actions]

Claimed by @AndriiPasternak31. Label status-in-progress added.

Next steps:

1. Create branch: feature/1444-<short-slug>
2. Implement the feature
3. Open a PR with Fixes #1444