Redundant id property for phase type messages is causing added latency for OpenAI models

[dileepyavan]

Tracking cleanup for Responses API prompt round-tripping.

Problem:
Assistant messages that participate in phase-based Responses API round-tripping are carrying a redundant standalone output message id in addition to the phase metadata. This creates avoidable payload overhead and additional processing for OpenAI model requests.

Desired behavior:

• Attach the response output message id only for assistant messages that have phase metadata.
• Store that id alongside the phase metadata rather than as a separate prompt payload.
• Send non-phase assistant history back as plain assistant input messages.
• Preserve phased assistant history as Responses API output messages with stable ids.

Acceptance criteria:

• Non-phase assistant messages do not include a response output message id in outgoing prompt history.
• Phased assistant messages retain the output message id only through phase metadata.
• OpenAI Responses API request construction and round-tripping tests cover this behavior.
• Existing compile/test validation remains clean.

[vs-code-engineering]

Hi @dileepyavan. As a member of the team, you can help us triage this issue by referring to https://github.com/microsoft/vscode-copilot/wiki/Copilot-Inbox-Tracker and assigning an owner directly.