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/codex.rs

@@ -184,7 +184,6 @@ impl Codex {
             base_instructions: config.base_instructions.clone(),
             approval_policy: config.approval_policy,
             sandbox_policy: config.sandbox_policy.clone(),
-            disable_response_storage: config.disable_response_storage,
             notify: config.notify.clone(),
             cwd: config.cwd.clone(),
         };
@@ -301,7 +300,6 @@ pub(crate) struct TurnContext {
     pub(crate) approval_policy: AskForApproval,
     pub(crate) sandbox_policy: SandboxPolicy,
     pub(crate) shell_environment_policy: ShellEnvironmentPolicy,
-    pub(crate) disable_response_storage: bool,
     pub(crate) tools_config: ToolsConfig,
 }
 
@@ -334,8 +332,6 @@ struct ConfigureSession {
     approval_policy: AskForApproval,
     /// How to sandbox commands executed in the system
     sandbox_policy: SandboxPolicy,
-    /// Disable server-side response storage (send full context each request)
-    disable_response_storage: bool,
 
     /// Optional external notifier command tokens. Present only when the
     /// client wants the agent to spawn a program after each completed
@@ -370,7 +366,6 @@ impl Session {
             base_instructions,
             approval_policy,
             sandbox_policy,
-            disable_response_storage,
             notify,
             cwd,
         } = configure_session;
@@ -462,7 +457,6 @@ impl Session {
             sandbox_policy,
             shell_environment_policy: config.shell_environment_policy.clone(),
             cwd,
-            disable_response_storage,
         };
         let sess = Arc::new(Session {
             session_id,
@@ -1117,7 +1111,6 @@ async fn submission_loop(
                     sandbox_policy: new_sandbox_policy.clone(),
                     shell_environment_policy: prev.shell_environment_policy.clone(),
                     cwd: new_cwd.clone(),
-                    disable_response_storage: prev.disable_response_storage,
                 };
 
                 // Install the new persistent context for subsequent tasks/turns.
@@ -1199,7 +1192,6 @@ async fn submission_loop(
                         sandbox_policy,
                         shell_environment_policy: turn_context.shell_environment_policy.clone(),
                         cwd,
-                        disable_response_storage: turn_context.disable_response_storage,
                     };
                     // TODO: record the new environment context in the conversation history
                     // no current task, spawn a new one with the per‑turn context
@@ -1604,7 +1596,6 @@ async fn run_turn(
 
     let prompt = Prompt {
         input,
-        store: !turn_context.disable_response_storage,
         tools,
         base_instructions_override: turn_context.base_instructions.clone(),
     };
@@ -1858,7 +1849,6 @@ async fn run_compact_task(
 
     let prompt = Prompt {
         input: turn_input,
-        store: !turn_context.disable_response_storage,
         tools: Vec::new(),
         base_instructions_override: Some(compact_instructions.clone()),
     };