feat: refactor CodexAuth so invalid state cannot be represented (#10208)

Previously, `CodexAuth` was defined as follows:


d550fbf41a/codex-rs/core/src/auth.rs (L39-L46)

But if you looked at its constructors, we had creation for
`AuthMode::ApiKey` where `storage` was built using a nonsensical path
(`PathBuf::new()`) and `auth_dot_json` was `None`:


d550fbf41a/codex-rs/core/src/auth.rs (L212-L220)

By comparison, when `AuthMode::ChatGPT` was used, `api_key` was always
`None`:


d550fbf41a/codex-rs/core/src/auth.rs (L665-L671)

https://github.com/openai/codex/pull/10012 took things further because
it introduced a new `ChatgptAuthTokens` variant to `AuthMode`, which is
important in when invoking `account/login/start` via the app server, but
most logic _internal_ to the app server should just reason about two
`AuthMode` variants: `ApiKey` and `ChatGPT`.

This PR tries to clean things up as follows:

- `LoginAccountParams` and `AuthMode` in `codex-rs/app-server-protocol/`
both continue to have the `ChatgptAuthTokens` variant, though it is used
exclusively for the on-the-wire messaging.
- `codex-rs/core/src/auth.rs` now has its own `AuthMode` enum, which
only has two variants: `ApiKey` and `ChatGPT`.
- `CodexAuth` has been changed from a struct to an enum. It is a
disjoint union where each variant (`ApiKey`, `ChatGpt`, and
`ChatGptAuthTokens`) have only the associated fields that make sense for
that variant.

---
[//]: # (BEGIN SAPLING FOOTER)
Stack created with [Sapling](https://sapling-scm.com). Best reviewed
with [ReviewStack](https://reviewstack.dev/openai/codex/pull/10208).
* #10224
* __->__ #10208

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

@@ -133,6 +133,7 @@ use codex_app_server_protocol::build_turns_from_event_msgs;
 use codex_backend_client::Client as BackendClient;
 use codex_chatgpt::connectors;
 use codex_core::AuthManager;
+use codex_core::CodexAuth;
 use codex_core::CodexThread;
 use codex_core::Cursor as RolloutCursor;
 use codex_core::InitialHistory;
@@ -686,7 +687,11 @@ impl CodexMessageProcessor {
                     .await;
 
                 let payload = AuthStatusChangeNotification {
-                    auth_method: self.auth_manager.auth_cached().map(|auth| auth.mode),
+                    auth_method: self
+                        .auth_manager
+                        .auth_cached()
+                        .as_ref()
+                        .map(CodexAuth::api_auth_mode),
                 };
                 self.outgoing
                     .send_server_notification(ServerNotification::AuthStatusChange(payload))
@@ -716,7 +721,11 @@ impl CodexMessageProcessor {
                     .await;
 
                 let payload_v2 = AccountUpdatedNotification {
-                    auth_mode: self.auth_manager.auth_cached().map(|auth| auth.mode),
+                    auth_mode: self
+                        .auth_manager
+                        .auth_cached()
+                        .as_ref()
+                        .map(CodexAuth::api_auth_mode),
                 };
                 self.outgoing
                     .send_server_notification(ServerNotification::AccountUpdated(payload_v2))
@@ -812,7 +821,10 @@ impl CodexMessageProcessor {
                             auth_manager.reload();
 
                             // Notify clients with the actual current auth mode.
-                            let current_auth_method = auth_manager.auth_cached().map(|a| a.mode);
+                            let current_auth_method = auth_manager
+                                .auth_cached()
+                                .as_ref()
+                                .map(CodexAuth::api_auth_mode);
                             let payload = AuthStatusChangeNotification {
                                 auth_method: current_auth_method,
                             };
@@ -902,7 +914,10 @@ impl CodexMessageProcessor {
                             auth_manager.reload();
 
                             // Notify clients with the actual current auth mode.
-                            let current_auth_method = auth_manager.auth_cached().map(|a| a.mode);
+                            let current_auth_method = auth_manager
+                                .auth_cached()
+                                .as_ref()
+                                .map(CodexAuth::api_auth_mode);
                             let payload_v2 = AccountUpdatedNotification {
                                 auth_mode: current_auth_method,
                             };
@@ -1106,7 +1121,11 @@ impl CodexMessageProcessor {
         }
 
         // Reflect the current auth method after logout (likely None).
-        Ok(self.auth_manager.auth_cached().map(|auth| auth.mode))
+        Ok(self
+            .auth_manager
+            .auth_cached()
+            .as_ref()
+            .map(CodexAuth::api_auth_mode))
     }
 
     async fn logout_v1(&mut self, request_id: RequestId) {
@@ -1178,7 +1197,7 @@ impl CodexMessageProcessor {
         } else {
             match self.auth_manager.auth().await {
                 Some(auth) => {
-                    let auth_mode = auth.mode;
+                    let auth_mode = auth.api_auth_mode();
                     let (reported_auth_method, token_opt) = match auth.get_token() {
                         Ok(token) if !token.is_empty() => {
                             let tok = if include_token { Some(token) } else { None };
@@ -1225,9 +1244,9 @@ impl CodexMessageProcessor {
         }
 
         let account = match self.auth_manager.auth_cached() {
-            Some(auth) => Some(match auth.mode {
-                AuthMode::ApiKey => Account::ApiKey {},
-                AuthMode::ChatGPT | AuthMode::ChatgptAuthTokens => {
+            Some(auth) => Some(match auth {
+                CodexAuth::ApiKey(_) => Account::ApiKey {},
+                CodexAuth::ChatGpt(_) | CodexAuth::ChatGptAuthTokens(_) => {
                     let email = auth.get_account_email();
                     let plan_type = auth.account_plan_type();
 
@@ -1286,7 +1305,7 @@ impl CodexMessageProcessor {
             });
         };
 
-        if !matches!(auth.mode, AuthMode::ChatGPT | AuthMode::ChatgptAuthTokens) {
+        if !auth.is_chatgpt_auth() {
             return Err(JSONRPCErrorError {
                 code: INVALID_REQUEST_ERROR_CODE,
                 message: "chatgpt authentication required to read rate limits".to_string(),