Never store requests (#3212)

When item ids are sent to Responses API it will load them from the
database ignoring the provided values. This adds extra latency.

Not having the mode to store requests also allows us to simplify the
code.

## Breaking change

The `disable_response_storage` configuration option is removed.

codex-rs/core/src/client.rs

@@ -157,14 +157,6 @@ impl ModelClient {
 
         let auth_manager = self.auth_manager.clone();
 
-        let auth_mode = auth_manager
-            .as_ref()
-            .and_then(|m| m.auth())
-            .as_ref()
-            .map(|a| a.mode);
-
-        let store = prompt.store && auth_mode != Some(AuthMode::ChatGPT);
-
         let full_instructions = prompt.get_full_instructions(&self.config.model_family);
         let tools_json = create_tools_json_for_responses_api(&prompt.tools)?;
         let reasoning = create_reasoning_param_for_request(
@@ -173,9 +165,7 @@ impl ModelClient {
             self.summary,
         );
 
-        // Request encrypted COT if we are not storing responses,
-        // otherwise reasoning items will be referenced by ID
-        let include: Vec<String> = if !store && reasoning.is_some() {
+        let include: Vec<String> = if reasoning.is_some() {
             vec!["reasoning.encrypted_content".to_string()]
         } else {
             vec![]
@@ -204,7 +194,7 @@ impl ModelClient {
             tool_choice: "auto",
             parallel_tool_calls: false,
             reasoning,
-            store,
+            store: false,
             stream: true,
             include,
             prompt_cache_key: Some(self.session_id.to_string()),