[App-server] Implement v2 for account/login/start and account/login/completed (#6183)

This PR implements `account/login/start` and `account/login/completed`.
Instead of having separate endpoints for login with chatgpt and api, we
have a single enum handling different login methods. For sync auth
methods like sign in with api key, we still send a `completed`
notification back to be compatible with the async login flow.

codex-rs/app-server/tests/common/mcp_process.rs

@@ -366,6 +366,26 @@ impl McpProcess {
         self.send_request("account/logout", None).await
     }
 
+    /// Send an `account/login/start` JSON-RPC request for API key login.
+    pub async fn send_login_account_api_key_request(
+        &mut self,
+        api_key: &str,
+    ) -> anyhow::Result<i64> {
+        let params = serde_json::json!({
+            "type": "apiKey",
+            "apiKey": api_key,
+        });
+        self.send_request("account/login/start", Some(params)).await
+    }
+
+    /// Send an `account/login/start` JSON-RPC request for ChatGPT login.
+    pub async fn send_login_account_chatgpt_request(&mut self) -> anyhow::Result<i64> {
+        let params = serde_json::json!({
+            "type": "chatgpt"
+        });
+        self.send_request("account/login/start", Some(params)).await
+    }
+
     /// Send a `fuzzyFileSearch` JSON-RPC request.
     pub async fn send_fuzzy_file_search_request(
         &mut self,