The /v1/responses endpoint generated a new UUID session_id for every
request, even when previous_response_id was provided. This caused each
turn of a multi-turn conversation to appear as a separate session on the
web dashboard, despite the conversation history being correctly chained.
Fix: store session_id alongside the response in the ResponseStore, and
reuse it when a subsequent request chains via previous_response_id.
Applies to both the non-streaming /v1/responses path and the streaming
SSE path. The /v1/runs endpoint also gains session continuity from
stored responses (explicit body.session_id still takes priority).
Adds test verifying session_id is preserved across chained requests.
5cbb45d93e