Remove redundant Codex Apps cache guard (#29575)

## Why

Codex Apps cache writes are already restricted to Codex Apps call paths:
startup invokes the helper only from the Codex Apps branch, and hard
refresh operates on the reserved Codex Apps server directly. Rechecking
the server name inside the cache helper duplicates that classification
and leaves the helper with an argument that cannot change valid
behavior.

## What changed

- Remove the redundant server-name check and parameter from the cache
writer.
- Rename the helper to `write_codex_apps_tools_cache` to reflect its
narrower contract.
- Update production and test callsites to use the simplified API.

codex-rs/codex-mcp/src/connection_manager.rs

@@ -16,7 +16,7 @@ use std::time::Instant;
 use crate::McpAuthStatusEntry;
 use crate::codex_apps::CodexAppsToolsCacheContext;
 use crate::codex_apps::CodexAppsToolsCacheKey;
-use crate::codex_apps::write_cached_codex_apps_tools_if_needed;
+use crate::codex_apps::write_codex_apps_tools_cache;
 use crate::elicitation::ElicitationRequestManager;
 use crate::elicitation::ElicitationReviewerHandle;
 use crate::mcp::CODEX_APPS_MCP_SERVER_NAME;
@@ -521,8 +521,7 @@ impl McpConnectionManager {
             &[],
         );
 
-        write_cached_codex_apps_tools_if_needed(
-            CODEX_APPS_MCP_SERVER_NAME,
+        write_codex_apps_tools_cache(
             managed_client.codex_apps_tools_cache_context.as_ref(),
             &managed_client.server_info,
             &tools,