[stack 2/4] Align main realtime v2 wire and runtime flow (#14830)

## Stack Position
2/4. Built on top of #14828.

## Base
- #14828

## Unblocks
- #14829
- #14827

## Scope
- Port the realtime v2 wire parsing, session, app-server, and
conversation runtime behavior onto the split websocket-method base.
- Branch runtime behavior directly on the current realtime session kind
instead of parser-derived flow flags.
- Keep regression coverage in the existing e2e suites.

---------

Co-authored-by: Codex <noreply@openai.com>

codex-rs/core/src/codex.rs

@@ -2614,6 +2614,9 @@ impl Session {
         if !matches!(msg, EventMsg::TurnComplete(_)) {
             return;
         }
+        if let Err(err) = self.conversation.handoff_complete().await {
+            debug!("failed to finalize realtime handoff output: {err}");
+        }
         self.conversation.clear_active_handoff().await;
     }