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/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",