Refactor handler context to be transport- and handler-type-aware

The current RequestContext / ServerRequestContext design has a few rough edges that warrant a dedicated refactor:

1. Request vs notification context — Request handlers have request_id and meta, notification handlers do not. Currently request_id is optional (RequestId | None) on a single shared type, which loses type safety. Ideally these would be distinct types so request handlers get a context where request_id is always present, and notification handlers get one where it does not exist.

2. Transport-specific context — The request_context field on ServerMessageMetadata is typed as Any because the server layer is transport-agnostic (e.g. it is a Starlette Request for HTTP transports, None for stdio). A better design would let transports provide their own typed context that handlers can access without casting.

These two concerns are related — both point toward making the context type more precise depending on how and where a handler is invoked.

_{AI Disclaimer}

首先沿著 handler 入口點追蹤 RequestContext 和 ServerRequestContext，然後檢查 ServerMessageMetadata.request_context，以及針對 Starlette Request 和 stdio 提到的傳輸情況。比較 request handler 和 notification handler 如何接收 context，以及如何存取傳輸特定的值。在不依賴 Any 或可選 request ID 的情況下定義預期的型別邊界和傳輸行為，即表示完成。