[Desktop] Show effective model in background-agent list and detail views

[FBakkensen]

This was generated by AI during triage.

Problem

Desktop Background agents list and detail views do not show which model performed the delegated work.

Task names sometimes mention a model, but they are user-authored and are not reliable metadata. Runtime model metadata is already persisted per session and usage event.

Motivation

Model visibility helps users audit delegated work, compare agent runs, and understand Auto routing or differences in output.

Proposed solution

Enrich the existing Desktop surfaces:

• Background agents list: show a short effective-model badge.
• Agent detail/header: show the model display name and stable model ID.
• Support truncation and tooltips; do not rely on color alone.

Display semantics

• Running: latest effective model.
• Completed: effective model used.
• Auto: concrete routed model, with configured Auto optionally shown secondarily.
• Unavailable: Unknown; never infer from the parent session.
• Multiple models: Mixed in the list, with models shown per turn or as a list in details.

Terminology should align with the effective-model states introduced by #3824.

Example workflow

1. Delegate several background agents using pinned and Auto-selected models.
2. Open Background agents to compare the effective models.
3. Open an agent to see its full model identity.

Acceptance criteria

• List rows show the effective model when available.
• Detail headers show the display name and stable model ID.
• Auto-routed agents show the concrete effective model.
• Missing metadata displays Unknown.
• Multi-model agents display Mixed, with details available in the agent view.
• A live Resolving... transition is optional.

Out of scope

• A new CLI/TUI worker panel.
• Reasoning-effort or context-tier display.
• Changing model selection or routing.
• Making the subagent aware of its model.

Related issues

• Collapsible worker/background agent panel showing active sub-agents, model, reasoning level, and task description #1783 - requests a new CLI/TUI worker panel; this request concerns existing Desktop views.
• Sub-agents can run a different model than the configured session model (agent-type defaults + experiment override), with no surfacing #3824 - effective-model display in the CLI; useful parity context.

Environment

• GitHub Copilot desktop app v1.0.25
• App-spawned CLI v1.0.71
• Windows 11

Cropped screenshots can be attached showing the missing model field in the Background agents list and agent header.