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/protocol/src/protocol.rs

@@ -190,6 +190,8 @@ pub struct ConversationStartParams {
     pub model: Option<String>,
     /// Selects whether the realtime session should produce text or audio output.
     pub output_modality: RealtimeOutputModality,
+    /// Whether to append Codex's startup context to the realtime backend prompt.
+    pub include_startup_context: bool,
     pub prompt: Option<Option<String>>,
     pub realtime_session_id: Option<String>,
     pub transport: Option<ConversationStartTransport>,