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/core/tests/common/test_codex.rs

@@ -641,6 +641,7 @@ impl TestCodexBuilder {
                         thread_source: None,
                         dynamic_tools: Vec::new(),
                         metrics_service_name: None,
+                        multi_agent_mode: None,
                         parent_trace: None,
                         environments,
                         thread_extension_init: Default::default(),

codex-rs/core/tests/suite/agents_md.rs

@@ -441,6 +441,7 @@ async fn loads_user_instructions_without_a_primary_environment() -> Result<()> {
             thread_source: None,
             dynamic_tools: Vec::new(),
             metrics_service_name: None,
+            multi_agent_mode: None,
             parent_trace: None,
             environments: Vec::new(),
             thread_extension_init: Default::default(),
@@ -648,6 +649,7 @@ async fn multi_environment_thread_loads_every_project_and_keeps_creation_snapsho
             thread_source: None,
             dynamic_tools: Vec::new(),
             metrics_service_name: None,
+            multi_agent_mode: None,
             parent_trace: None,
             environments: vec![
                 TurnEnvironmentSelection {

codex-rs/core/tests/suite/subagent_notifications.rs

@@ -751,6 +751,7 @@ async fn subagent_stop_replaces_stop_and_skips_internal_subagents() -> Result<()
             thread_source: None,
             dynamic_tools: Vec::new(),
             metrics_service_name: None,
+            multi_agent_mode: None,
             parent_trace: None,
             environments: Vec::new(),
             thread_extension_init: Default::default(),