[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/app-server/tests/suite/v2/realtime_conversation.rs

@@ -70,6 +70,7 @@ async fn realtime_conversation_streams_v2_notifications() -> Result<()> {
                 "message": "upstream boom"
             }),
         ],
+        vec![],
     ]])
     .await;
 
@@ -135,6 +136,7 @@ async fn realtime_conversation_streams_v2_notifications() -> Result<()> {
                 sample_rate: 24_000,
                 num_channels: 1,
                 samples_per_channel: Some(480),
+                item_id: None,
             },
         })
         .await?;
@@ -191,7 +193,7 @@ async fn realtime_conversation_streams_v2_notifications() -> Result<()> {
     let connections = realtime_server.connections();
     assert_eq!(connections.len(), 1);
     let connection = &connections[0];
-    assert_eq!(connection.len(), 3);
+    assert_eq!(connection.len(), 4);
     assert_eq!(
         connection[0].body_json()["type"].as_str(),
         Some("session.update")
@@ -211,6 +213,10 @@ async fn realtime_conversation_streams_v2_notifications() -> Result<()> {
             .as_str()
             .context("expected websocket request type")?
             .to_string(),
+        connection[3].body_json()["type"]
+            .as_str()
+            .context("expected websocket request type")?
+            .to_string(),
     ];
     request_types.sort();
     assert_eq!(
@@ -218,6 +224,7 @@ async fn realtime_conversation_streams_v2_notifications() -> Result<()> {
         [
             "conversation.item.create".to_string(),
             "input_audio_buffer.append".to_string(),
+            "response.create".to_string(),
         ]
     );