feat: expose outputSchema to user_turn/turn_start app_server API (#8377)

What changed
- Added `outputSchema` support to the app-server APIs, mirroring `codex
exec --output-schema` behavior.
- V1 `sendUserTurn` now accepts `outputSchema` and constrains the final
assistant message for that turn.
- V2 `turn/start` now accepts `outputSchema` and constrains the final
assistant message for that turn (explicitly per-turn only).

Core behavior
- `Op::UserTurn` already supported `final_output_json_schema`; now V1
`sendUserTurn` forwards `outputSchema` into that field.
- `Op::UserInput` now carries `final_output_json_schema` for per-turn
settings updates; core maps it into
`SessionSettingsUpdate.final_output_json_schema` so it applies to the
created turn context.
- V2 `turn/start` does NOT persist the schema via `OverrideTurnContext`
(it’s applied only for the current turn). Other overrides
(cwd/model/etc) keep their existing persistent behavior.

API / docs
- `codex-rs/app-server-protocol/src/protocol/v1.rs`: add `output_schema:
Option<serde_json::Value>` to `SendUserTurnParams` (serialized as
`outputSchema`).
- `codex-rs/app-server-protocol/src/protocol/v2.rs`: add `output_schema:
Option<JsonValue>` to `TurnStartParams` (serialized as `outputSchema`).
- `codex-rs/app-server/README.md`: document `outputSchema` for
`turn/start` and clarify it applies only to the current turn.
- `codex-rs/docs/codex_mcp_interface.md`: document `outputSchema` for v1
`sendUserTurn` and v2 `turn/start`.

Tests added/updated
- New app-server integration tests asserting `outputSchema` is forwarded
into outbound `/responses` requests as `text.format`:
  - `codex-rs/app-server/tests/suite/output_schema.rs`
  - `codex-rs/app-server/tests/suite/v2/output_schema.rs`
- Added per-turn semantics tests (schema does not leak to the next
turn):
  - `send_user_turn_output_schema_is_per_turn_v1`
  - `turn_start_output_schema_is_per_turn_v2`
- Added protocol wire-compat tests for the merged op:
  - serialize omits `final_output_json_schema` when `None`
  - deserialize works when field is missing
  - serialize includes `final_output_json_schema` when `Some(schema)`

Call site updates (high level)
- Updated all `Op::UserInput { .. }` constructions to include
`final_output_json_schema`:
  - `codex-rs/app-server/src/codex_message_processor.rs`
  - `codex-rs/core/src/codex_delegate.rs`
  - `codex-rs/mcp-server/src/codex_tool_runner.rs`
  - `codex-rs/tui/src/chatwidget.rs`
  - `codex-rs/tui2/src/chatwidget.rs`
  - plus impacted core tests.

Validation
- `just fmt`
- `cargo test -p codex-core`
- `cargo test -p codex-app-server`
- `cargo test -p codex-mcp-server`
- `cargo test -p codex-tui`
- `cargo test -p codex-tui2`
- `cargo test -p codex-protocol`
- `cargo clippy --all-features --tests --profile dev --fix -- -D
warnings`

codex-rs/app-server/README.md

@@ -162,7 +162,7 @@ Turns attach user input (text or images) to a thread and trigger Codex generatio
 - `{"type":"image","url":"https://…png"}`
 - `{"type":"localImage","path":"/tmp/screenshot.png"}`
 
-You can optionally specify config overrides on the new turn. If specified, these settings become the default for subsequent turns on the same thread.
+You can optionally specify config overrides on the new turn. If specified, these settings become the default for subsequent turns on the same thread. `outputSchema` applies only to the current turn.
 
 ```json
 { "method": "turn/start", "id": 30, "params": {
@@ -178,7 +178,14 @@ You can optionally specify config overrides on the new turn. If specified, these
     },
     "model": "gpt-5.1-codex",
     "effort": "medium",
-    "summary": "concise"
+    "summary": "concise",
+    // Optional JSON Schema to constrain the final assistant message for this turn.
+    "outputSchema": {
+        "type": "object",
+        "properties": { "answer": { "type": "string" } },
+        "required": ["answer"],
+        "additionalProperties": false
+    }
 } }
 { "id": 30, "result": { "turn": {
     "id": "turn_456",