Send base instructions as Responses input messages (#51156)

## What changed

Move base instructions from the top-level `instructions` field into a prefixed `developer` message in `input` for standard Responses requests, matching Responses Lite. Remove the field from HTTP and WebSocket request types and assign instruction messages stable IDs derived from the thread and instruction text.

## Testing

Extend standard and Responses Lite tests to cover instruction message serialization, stable IDs, and resume consistency. Update WebSocket tests to verify that changed instructions trigger a full request without `previous_response_id`.

GitOrigin-RevId: e2de4b8e30a9339377fac133ceda981abd1a112b

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

@@ -283,8 +283,6 @@ pub struct ResponsesApiRequest {
     pub stream: bool,
     #[serde(skip_serializing_if = "Option::is_none")]
     pub service_tier: Option<String>,
-    #[serde(skip_serializing_if = "String::is_empty")]
-    pub instructions: String,
     pub input: Vec<ResponseItem>,
     #[serde(skip_serializing_if = "Option::is_none")]
     pub tools: Option<ResponsesApiTools>,
@@ -309,7 +307,6 @@ impl<'a> From<&'a ResponsesApiRequest> for ResponseCreateWsRequest<'a> {
     fn from(request: &'a ResponsesApiRequest) -> Self {
         Self {
             model: &request.model,
-            instructions: &request.instructions,
             previous_response_id: None,
             input: &request.input,
             tools: request.tools.as_ref().map(ResponsesApiTools::as_raw_value),
@@ -338,8 +335,6 @@ pub struct ResponseCreateWsRequest<'a> {
     pub stream: bool,
     #[serde(skip_serializing_if = "Option::is_none")]
     pub service_tier: Option<&'a str>,
-    #[serde(skip_serializing_if = "str::is_empty")]
-    pub instructions: &'a str,
     #[serde(skip_serializing_if = "Option::is_none")]
     pub previous_response_id: Option<String>,
     pub input: &'a [ResponseItem],