Disable storage for Azure Responses requests (#38060)

## What changed

- Set `store` to `false` for every Responses request, including requests sent
  through Azure providers.
- Remove the provider-specific storage check and simplify request construction.
- Update the Azure request test to verify that storage is disabled while
  prefixed item IDs remain unchanged.

GitOrigin-RevId: 1c71cb80880058cb26789112f03269b73f0645b9

codex-rs/core/src/client.rs

@@ -577,7 +577,6 @@ impl ModelClient {
             self.state.auth_env_telemetry.clone(),
         );
         let request = self.build_responses_request(
-            &client_setup.api_provider,
             prompt,
             model_info,
             settings.effort,
@@ -842,10 +841,8 @@ impl ModelClient {
         }
     }
 
-    #[allow(clippy::too_many_arguments)]
     fn build_responses_request(
         &self,
-        provider: &codex_api::Provider,
         prompt: &Prompt,
         model_info: &ModelInfo,
         effort: Option<ReasoningEffortConfig>,
@@ -931,7 +928,7 @@ impl ModelClient {
             tool_choice: "auto".to_string(),
             parallel_tool_calls: prompt.parallel_tool_calls && !model_info.use_responses_lite,
             reasoning: Some(reasoning),
-            store: provider.is_azure_responses_endpoint(),
+            store: false,
             stream: true,
             stream_options,
             include,
@@ -1482,7 +1479,6 @@ impl ModelClientSession {
                 .await;
 
             let mut request = self.client.build_responses_request(
-                &client_setup.api_provider,
                 prompt,
                 model_info,
                 effort.clone(),
@@ -1603,7 +1599,6 @@ impl ModelClientSession {
                 pending_retry,
             );
             let mut request = self.client.build_responses_request(
-                &client_setup.api_provider,
                 prompt,
                 model_info,
                 effort.clone(),