add WebSearchMode enum (#9216)

### What
Add `WebSearchMode` enum (disabled, cached live, defaults to cached) to
config + V2 protocol. This enum takes precedence over legacy flags:
`web_search_cached`, `web_search_request`, and `tools.web_search`.

Keep `--search` as live.

### Tests
Added tests

codex-rs/core/config.schema.json

@@ -395,6 +395,14 @@
         }
       ]
     },
+    "web_search": {
+      "description": "Controls the web search tool mode: disabled, cached, or live.",
+      "allOf": [
+        {
+          "$ref": "#/definitions/WebSearchMode"
+        }
+      ]
+    },
     "windows_wsl_setup_acknowledged": {
       "description": "Tracks whether the Windows onboarding screen has been acknowledged.",
       "type": "boolean"
@@ -634,6 +642,9 @@
         },
         "tools_web_search": {
           "type": "boolean"
+        },
+        "web_search": {
+          "$ref": "#/definitions/WebSearchMode"
         }
       },
       "additionalProperties": false
@@ -1420,6 +1431,14 @@
         "high"
       ]
     },
+    "WebSearchMode": {
+      "type": "string",
+      "enum": [
+        "disabled",
+        "cached",
+        "live"
+      ]
+    },
     "WireApi": {
       "description": "Wire protocol that the provider speaks. Most third-party services only implement the classic OpenAI Chat Completions JSON schema, whereas OpenAI itself (and a handful of others) additionally expose the more modern *Responses* API. The two protocols use different request/response shapes and *cannot* be auto-detected at runtime, therefore each provider entry must declare which one it expects.",
       "oneOf": [