feat(app-server, core): add more spans (#14479)

## Description

This PR expands tracing coverage across app-server thread startup, core
session initialization, and the Responses transport layer. It also gives
core dispatch spans stable operation-specific names so traces are easier
to follow than the old generic `submission_dispatch` spans.

Also use `fmt::Display` for types that we serialize in traces so we send
strings instead of rust types

codex-rs/codex-api/src/endpoint/responses.rs

@@ -21,6 +21,7 @@ use http::Method;
 use serde_json::Value;
 use std::sync::Arc;
 use std::sync::OnceLock;
+use tracing::instrument;
 
 pub struct ResponsesClient<T: HttpTransport, A: AuthProvider> {
     session: EndpointSession<T, A>,
@@ -55,6 +56,16 @@ impl<T: HttpTransport, A: AuthProvider> ResponsesClient<T, A> {
         }
     }
 
+    #[instrument(
+        name = "responses.stream_request",
+        level = "info",
+        skip_all,
+        fields(
+            transport = "responses_http",
+            http.method = "POST",
+            api.path = "responses"
+        )
+    )]
     pub async fn stream_request(
         &self,
         request: ResponsesApiRequest,
@@ -90,6 +101,17 @@ impl<T: HttpTransport, A: AuthProvider> ResponsesClient<T, A> {
         "responses"
     }
 
+    #[instrument(
+        name = "responses.stream",
+        level = "info",
+        skip_all,
+        fields(
+            transport = "responses_http",
+            http.method = "POST",
+            api.path = "responses",
+            turn.has_state = turn_state.is_some()
+        )
+    )]
     pub async fn stream(
         &self,
         body: Value,