Expose thread-level multi-agent mode (#28792)

## Why

Once multi-agent mode can be selected per turn, clients also need to
choose the initial selection when creating a thread and observe that
selection through lifecycle and settings APIs.

The selected value is intentionally distinct from the effective
model-visible value: no client selection is represented as `null`, even
though an eligible multi-agent v2 turn derives `explicitRequestOnly` as
its effective default.

## What changed

- Add the optional experimental `thread/start.multiAgentMode` parameter
and pass it through thread creation.
- Preserve an omitted initial value as an unset selection rather than
eagerly storing `explicitRequestOnly`.
- Apply an explicit `thread/start` selection to the first turn through
the session configuration established at thread creation.
- Restore the latest persisted effective mode as the selected baseline
on cold resume when rollout history contains one.
- Inherit the optional selected mode from a loaded parent when creating
related runtime threads.
- Return the current selected `multiAgentMode` from `thread/start`,
`thread/resume`, `thread/fork`, and thread settings, using `null` when
no mode is selected.
- Keep lifecycle reporting independent from model capability and feature
eligibility; core turn construction remains responsible for calculating
and persisting the effective mode.

## Not covered

- Clearing an existing loaded-session selection back to unset through
`turn/start`; omitted or `null` currently retains the session's
selection.
- A TUI control, slash command, or `config.toml` preference.

## Verification

- `CARGO_INCREMENTAL=0 just test -p codex-app-server-protocol`
- `CARGO_INCREMENTAL=0 just test -p codex-app-server multi_agent_mode`

The focused app-server coverage verifies explicit `thread/start`
initialization, first-turn prompting, nullable reporting for an omitted
selection, and retention of selections that are not currently
runtime-eligible.

## Stack

Stacked on #28685. This PR contains only the thread initialization and
lifecycle/settings API layer.

codex-rs/tui/src/app/app_server_event_targets.rs

@@ -230,6 +230,7 @@ mod tests {
                     developer_instructions: None,
                 },
             },
+            multi_agent_mode: Default::default(),
             personality: None,
         }
     }

codex-rs/tui/src/app/tests.rs

@@ -6038,6 +6038,7 @@ async fn inactive_thread_settings_notification_updates_cached_collaboration_mode
             effort: collaboration_mode.settings.reasoning_effort.clone(),
             summary: None,
             collaboration_mode: collaboration_mode.clone(),
+            multi_agent_mode: Default::default(),
             personality: Some(Personality::Pragmatic),
         },
     };

codex-rs/tui/src/app_server_session.rs

@@ -2381,6 +2381,7 @@ mod tests {
                 .into(),
             active_permission_profile: None,
             reasoning_effort: None,
+            multi_agent_mode: Default::default(),
             initial_turns_page: None,
         };
 

codex-rs/tui/src/chatwidget/tests/app_server.rs

@@ -30,6 +30,7 @@ fn thread_settings_for_test(
                     developer_instructions: None,
                 },
             },
+            multi_agent_mode: Default::default(),
             personality: Some(Personality::Pragmatic),
         },
     }