fix: make ShutdownHandle a private field of LoginServer (#2396)

Folds the top-level `shutdown()` function into a method of
`ShutdownHandle` and then simply stores `ShutdownHandle` on
`LoginServer` since the two fields it contains were always being used
together, anyway.

---
[//]: # (BEGIN SAPLING FOOTER)
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with [ReviewStack](https://reviewstack.dev/openai/codex/pull/2396).
* #2399
* #2398
* __->__ #2396
* #2395
* #2394
* #2393
* #2389

codex-rs/mcp-server/src/codex_message_processor.rs

@@ -66,7 +66,7 @@ struct ActiveLogin {
 
 impl ActiveLogin {
     fn drop(&self) {
-        self.shutdown_handle.cancel();
+        self.shutdown_handle.shutdown();
     }
 }
 
@@ -190,7 +190,7 @@ impl CodexMessageProcessor {
                         Ok(Err(err)) => (false, Some(format!("Login server error: {err}"))),
                         Err(_elapsed) => {
                             // Timeout: cancel server and report
-                            shutdown_handle.cancel();
+                            shutdown_handle.shutdown();
                             (false, Some("Login timed out".to_string()))
                         }
                     };