[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/codex-api/tests/realtime_websocket_e2e.rs

@@ -170,6 +170,7 @@ async fn realtime_ws_e2e_session_create_and_event_flow() {
             sample_rate: 48000,
             num_channels: 1,
             samples_per_channel: Some(960),
+            item_id: None,
         })
         .await
         .expect("send audio");
@@ -186,6 +187,7 @@ async fn realtime_ws_e2e_session_create_and_event_flow() {
             sample_rate: 48000,
             num_channels: 1,
             samples_per_channel: None,
+            item_id: None,
         })
     );
 
@@ -254,6 +256,7 @@ async fn realtime_ws_e2e_send_while_next_event_waits() {
                     sample_rate: 48000,
                     num_channels: 1,
                     samples_per_channel: Some(960),
+                    item_id: None,
                 }),
             )
             .await