[app-server] update doc with codex error info (#6941)

Document new codex error info. Also fixed the name from
`codex_error_code` to `codex_error_info`.

codex-rs/app-server/README.md

@@ -339,6 +339,29 @@ Event notifications are the server-initiated event stream for thread lifecycles,
 
 The app-server streams JSON-RPC notifications while a turn is running. Each turn starts with `turn/started` (initial `turn`) and ends with `turn/completed` (final `turn` plus token `usage`), and clients subscribe to the events they care about, rendering each item incrementally as updates arrive. The per-item lifecycle is always: `item/started` → zero or more item-specific deltas → `item/completed`.
 
+- `turn/started` — `{ turn }` with the turn id, empty `items`, and `status: "inProgress"`.
+- `turn/completed` — `{ turn }` where `turn.status` is `completed`, `interrupted`, or `failed`; failures carry `{ error: { message, codexErrorInfo? } }`.
+
+Today both notifications carry an empty `items` array even when item events were streamed; rely on `item/*` notifications for the canonical item list until this is fixed.
+
+#### Errors
+`error` event is emitted whenever the server hits an error mid-turn (for example, upstream model errors or quota limits). Carries the same `{ error: { message, codexErrorInfo? } }` payload as `turn.status: "failed"` and may precede that terminal notification.
+
+  `codexErrorInfo` maps to the `CodexErrorInfo` enum. Common values:
+  - `ContextWindowExceeded`
+  - `UsageLimitExceeded`
+  - `HttpConnectionFailed { httpStatusCode? }`: upstream HTTP failures including 4xx/5xx
+  - `ResponseStreamConnectionFailed { httpStatusCode? }`: failure to connect to the response SSE stream
+  - `ResponseStreamDisconnected { httpStatusCode? }`: disconnect of the response SSE stream in the middle of a turn before completion
+  - `ResponseTooManyFailedAttempts { httpStatusCode? }`
+  - `BadRequest`
+  - `Unauthorized`
+  - `SandboxError`
+  - `InternalServerError`
+  - `Other`: all unclassified errors
+
+When an upstream HTTP status is available (for example, from the Responses API or a provider), it is forwarded in `httpStatusCode` on the relevant `codexErrorInfo` variant.
+
 #### Thread items
 
 `ThreadItem` is the tagged union carried in turn responses and `item/*` notifications. Currently we support events for the following items: