[codex] Use input items for Responses Lite tools (#27946)

When using Responses Lite, we should all use `additional_tools` and a
developer item instead of the top level tools array & instructions
field. This keeps things 1-to-1.

Forced namespacing for _all_ tools will land in a following PR after
some coordination & fixes in Responses API (around collisions & return
items).

The goal is to eventually expand the scope of this to _all_ requests
from codex, but that will require larger coordination across providers &
slower rollout.

codex-rs/codex-api/src/common.rs

@@ -28,7 +28,8 @@ pub struct CompactionInput<'a> {
     pub input: &'a [ResponseItem],
     #[serde(skip_serializing_if = "str::is_empty")]
     pub instructions: &'a str,
-    pub tools: Vec<Value>,
+    #[serde(skip_serializing_if = "Option::is_none")]
+    pub tools: Option<Vec<Value>>,
     pub parallel_tool_calls: bool,
     #[serde(skip_serializing_if = "Option::is_none")]
     pub reasoning: Option<Reasoning>,
@@ -210,7 +211,8 @@ pub struct ResponsesApiRequest {
     #[serde(skip_serializing_if = "String::is_empty")]
     pub instructions: String,
     pub input: Vec<ResponseItem>,
-    pub tools: Vec<serde_json::Value>,
+    #[serde(skip_serializing_if = "Option::is_none")]
+    pub tools: Option<Vec<serde_json::Value>>,
     pub tool_choice: String,
     pub parallel_tool_calls: bool,
     pub reasoning: Option<Reasoning>,
@@ -258,7 +260,8 @@ pub struct ResponseCreateWsRequest {
     #[serde(skip_serializing_if = "Option::is_none")]
     pub previous_response_id: Option<String>,
     pub input: Vec<ResponseItem>,
-    pub tools: Vec<Value>,
+    #[serde(skip_serializing_if = "Option::is_none")]
+    pub tools: Option<Vec<Value>>,
     pub tool_choice: String,
     pub parallel_tool_calls: bool,
     pub reasoning: Option<Reasoning>,