[codex] Require model for standalone web search (#25131)

## Why

The standalone `/v1/alpha/search` request now requires a `model`, but
the `web.run` extension currently omits it.

Adds `model` to extension `ToolCall` invocation.

Follow-up to #23823.

## What changed

- Make `SearchRequest.model` required.
- Expose the effective per-turn model on extension tool calls and pass
it in standalone web-search requests.
- Assert the model is forwarded in the app-server round-trip test.

## Testing

- `just test -p codex-api -p codex-tools -p codex-web-search-extension
-p codex-memories-extension -p codex-goal-extension`
- `just test -p codex-core -E
'test(passes_turn_fields_and_scoped_turn_item_emitter_to_extension_call)'`
- `just test -p codex-app-server -E
'test(standalone_web_search_round_trips_encrypted_output)'`

codex-rs/codex-api/src/endpoint/search.rs

@@ -145,7 +145,7 @@ mod tests {
             .search(
                 &SearchRequest {
                     id: "search-session".to_string(),
-                    model: Some("gpt-test".to_string()),
+                    model: "gpt-test".to_string(),
                     reasoning: None,
                     input: Some(SearchInput::Items(vec![ResponseItem::Message {
                         id: None,

codex-rs/codex-api/src/search.rs

@@ -7,8 +7,7 @@ use serde::Serialize;
 #[derive(Debug, Clone, Serialize, PartialEq)]
 pub struct SearchRequest {
     pub id: String,
-    #[serde(skip_serializing_if = "Option::is_none")]
-    pub model: Option<String>,
+    pub model: String,
     #[serde(skip_serializing_if = "Option::is_none")]
     pub reasoning: Option<Reasoning>,
     #[serde(skip_serializing_if = "Option::is_none")]