Call models endpoint in models manager (#7616)

- Introduce `with_remote_overrides` and update
`refresh_available_models`
- Put `auth_manager` instead of `auth_mode` on `models_manager`
- Remove `ShellType` and `ReasoningLevel` to use already existing
structs

codex-rs/core/tests/chat_completions_payload.rs

@@ -1,6 +1,8 @@
 use std::sync::Arc;
 
 use codex_app_server_protocol::AuthMode;
+use codex_core::AuthManager;
+use codex_core::CodexAuth;
 use codex_core::ContentItem;
 use codex_core::LocalShellAction;
 use codex_core::LocalShellExecAction;
@@ -71,7 +73,8 @@ async fn run_request(input: Vec<ResponseItem>) -> Value {
     let config = Arc::new(config);
 
     let conversation_id = ConversationId::new();
-    let models_manager = Arc::new(ModelsManager::new(Some(AuthMode::ApiKey)));
+    let auth_manager = AuthManager::from_auth_for_testing(CodexAuth::from_api_key("Test API Key"));
+    let models_manager = Arc::new(ModelsManager::new(auth_manager));
     let model_family = models_manager.construct_model_family(&config.model, &config);
     let otel_event_manager = OtelEventManager::new(
         conversation_id,