📝 refine library async write docs

docs/api/library-async-logger-error.md

@@ -3,7 +3,7 @@ name: library-async-logger-error
 group: api
 category: facade
 update-time: 20260614
-description: Enqueue an error-level record through a LibraryAsyncLogger facade using the highest built-in severity shortcut and the repo's direct async call style.
+description: Enqueue an error-level record through a LibraryAsyncLogger facade by delegating to the wrapped async logger's error shortcut.
 key-word:
     - async
     - library
@@ -39,10 +39,11 @@ pub async fn[S] LibraryAsyncLogger::error(
 
 Detailed rules explaining key parameters and behaviors
 
-- This helper delegates to `log(Level::Error, ...)` on the wrapped async logger.
-- The record is still subject to patching, filtering, and overflow policy.
+- This helper delegates to `error(...)` on the wrapped async logger, which in turn uses `log(Level::Error, ...)`.
+- The record is still subject to stored shared context fields, patching, filtering, and overflow policy.
 - Error records represent the highest built-in severity in this async facade API.
 - Use this helper when a named error call is clearer than a raw `log(...)` call.
+- Async state helpers remain on the underlying `AsyncLogger[S]` and require `to_async_logger()` first.
 
 ### How to Use
 
@@ -69,6 +70,8 @@ logger.error(
 
 In this example, the error record carries structured context without falling back to the generic `log(...)` form.
 
+And any shared context fields already stored on the facade are still prepended before these per-call fields.
+
 ### Error Case
 
 e.g.:
@@ -76,8 +79,12 @@ e.g.:
 
 - If callers need to inspect worker failure rather than emit an error record, `has_failed()` and `last_error()` are the relevant APIs on the full async logger.
 
+- If callers need a per-call target override, they should use `log(...)` instead of this fixed-level shortcut.
+
 ### Notes
 
 1. Use this helper for high-severity async application failures.
 
 2. Emitting an error record is separate from the logger worker itself entering failure state.
+
+3. Use `to_async_logger()` first when later code needs queue or failure inspection rather than another write shortcut.