Add a toggle for realtime startup context (#28405)

## Summary
- Add `includeStartupContext` to realtime start requests so callers can
explicitly skip Codex startup context while keeping the backend prompt
- Thread the new flag through protocol types, request processing, and
realtime session config
- Update app-server docs and coverage for the new default and opt-out
behavior

## Testing
- Added protocol serialization coverage for `includeStartupContext`
- Added realtime integration coverage for starting a session with
startup context disabled

codex-rs/app-server-protocol/src/protocol/common.rs

@@ -3043,6 +3043,7 @@ mod tests {
                 thread_id: "thr_123".to_string(),
                 model: Some("realtime-treatment-model".to_string()),
                 output_modality: RealtimeOutputModality::Audio,
+                include_startup_context: Some(false),
                 prompt: Some(Some("You are on a call".to_string())),
                 realtime_session_id: Some("sess_456".to_string()),
                 transport: None,
@@ -3061,6 +3062,7 @@ mod tests {
                     "codexResponseItemPrefix": null,
                     "model": "realtime-treatment-model",
                     "outputModality": "audio",
+                    "includeStartupContext": false,
                     "prompt": "You are on a call",
                     "realtimeSessionId": "sess_456",
                     "transport": null,
@@ -3084,6 +3086,7 @@ mod tests {
                 thread_id: "thr_123".to_string(),
                 model: None,
                 output_modality: RealtimeOutputModality::Audio,
+                include_startup_context: None,
                 prompt: None,
                 realtime_session_id: None,
                 transport: None,
@@ -3102,6 +3105,7 @@ mod tests {
                     "codexResponseItemPrefix": null,
                     "model": null,
                     "outputModality": "audio",
+                    "includeStartupContext": null,
                     "realtimeSessionId": null,
                     "transport": null,
                     "version": null,
@@ -3120,6 +3124,7 @@ mod tests {
                 thread_id: "thr_123".to_string(),
                 model: None,
                 output_modality: RealtimeOutputModality::Audio,
+                include_startup_context: None,
                 prompt: Some(None),
                 realtime_session_id: None,
                 transport: None,
@@ -3138,6 +3143,7 @@ mod tests {
                     "codexResponseItemPrefix": null,
                     "model": null,
                     "outputModality": "audio",
+                    "includeStartupContext": null,
                     "prompt": null,
                     "realtimeSessionId": null,
                     "transport": null,
@@ -3322,6 +3328,7 @@ mod tests {
                 thread_id: "thr_123".to_string(),
                 model: None,
                 output_modality: RealtimeOutputModality::Audio,
+                include_startup_context: None,
                 prompt: Some(Some("You are on a call".to_string())),
                 realtime_session_id: None,
                 transport: None,

codex-rs/app-server-protocol/src/protocol/v2/realtime.rs

@@ -82,6 +82,9 @@ pub struct ThreadRealtimeStartParams {
     /// Selects text or audio output for the realtime session. Transport and voice stay
     /// independent so clients can choose how they connect separately from what the model emits.
     pub output_modality: RealtimeOutputModality,
+    /// Set to false to start without Codex's startup context. Omitted or null includes it.
+    #[ts(optional = nullable)]
+    pub include_startup_context: Option<bool>,
     #[serde(
         default,
         deserialize_with = "crate::protocol::serde_helpers::deserialize_double_option",