fix: handle all web_search actions and in progress invocations (#9960)

### Summary
- Parse all `web_search` tool actions (`search`, `find_in_page`,
`open_page`).
- Previously we only parsed + displayed `search`, which made the TUI
appear to pause when the other actions were being used.
- Show in progress `web_search` calls as `Searching the web`
  - Previously we only showed completed tool calls

<img width="308" height="149" alt="image"
src="https://github.com/user-attachments/assets/90a4e8ff-b06a-48ff-a282-b57b31121845"
/>

### Tests
Added + updated tests, tested locally

### Follow ups
Update VSCode extension to display these as well

codex-rs/protocol/src/items.rs

@@ -1,3 +1,4 @@
+use crate::models::WebSearchAction;
 use crate::protocol::AgentMessageEvent;
 use crate::protocol::AgentReasoningEvent;
 use crate::protocol::AgentReasoningRawContentEvent;
@@ -49,10 +50,11 @@ pub struct ReasoningItem {
     pub raw_content: Vec<String>,
 }
 
-#[derive(Debug, Clone, Deserialize, Serialize, TS, JsonSchema)]
+#[derive(Debug, Clone, Deserialize, Serialize, TS, JsonSchema, PartialEq)]
 pub struct WebSearchItem {
     pub id: String,
     pub query: String,
+    pub action: WebSearchAction,
 }
 
 impl UserMessageItem {
@@ -181,6 +183,7 @@ impl WebSearchItem {
         EventMsg::WebSearchEnd(WebSearchEndEvent {
             call_id: self.id.clone(),
             query: self.query.clone(),
+            action: self.action.clone(),
         })
     }
 }