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

Previously, `CodexAuth` was defined as follows:


https://github.com/openai/codex/blob/d550fbf41afc09d7d7b5ac813aea38de07b2a73f/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`:


https://github.com/openai/codex/blob/d550fbf41afc09d7d7b5ac813aea38de07b2a73f/codex-rs/core/src/auth.rs#L212-L220

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


https://github.com/openai/codex/blob/d550fbf41afc09d7d7b5ac813aea38de07b2a73f/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)
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with [ReviewStack](https://reviewstack.dev/openai/codex/pull/10208).
* #10224
* __->__ #10208

codex-rs/core/src/model_provider_info.rs

@@ -5,10 +5,11 @@
 //!   2. User-defined entries inside `~/.codex/config.toml` under the `model_providers`
 //!      key. These override or extend the defaults at runtime.
 
+use crate::auth::AuthMode;
+use crate::error::EnvVarError;
 use codex_api::Provider as ApiProvider;
 use codex_api::WireApi as ApiWireApi;
 use codex_api::provider::RetryConfig as ApiRetryConfig;
-use codex_app_server_protocol::AuthMode;
 use http::HeaderMap;
 use http::header::HeaderName;
 use http::header::HeaderValue;
@@ -19,7 +20,6 @@ use std::collections::HashMap;
 use std::env::VarError;
 use std::time::Duration;
 
-use crate::error::EnvVarError;
 const DEFAULT_STREAM_IDLE_TIMEOUT_MS: u64 = 300_000;
 const DEFAULT_STREAM_MAX_RETRIES: u64 = 5;
 const DEFAULT_REQUEST_MAX_RETRIES: u64 = 4;
@@ -137,10 +137,7 @@ impl ModelProviderInfo {
         &self,
         auth_mode: Option<AuthMode>,
     ) -> crate::error::Result<ApiProvider> {
-        let default_base_url = if matches!(
-            auth_mode,
-            Some(AuthMode::ChatGPT | AuthMode::ChatgptAuthTokens)
-        ) {
+        let default_base_url = if matches!(auth_mode, Some(AuthMode::ChatGPT)) {
             "https://chatgpt.com/backend-api/codex"
         } else {
             "https://api.openai.com/v1"