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

[maxisbey]

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}

[maxisbey]

Adding context from #2038 / #953 / #2001: the related_request_id pattern is a concrete example of the problem described in this issue.

Every Context method that sends a notification must manually thread related_request_id=self.request_id:

• report_progress() — was missing it, causing silent notification drops on streamable HTTP (the bug in context.report_progress not sending notifications on streamable-http #953/Progress notifications not delivered via SSE in stateless HTTP mode #2001, open for 8 months)
• log()
• elicit()
• elicit_url()

Same bug exists unfixed in:

• ProgressContext.progress() — no related_request_id
• TaskContext._send_notification() — potentially missing it

The fix in #2038 is a band-aid (adding the missing kwarg). The architectural fix would be to centralize related_request_id binding in Context (e.g., a request-scoped session wrapper or a send_notification helper) so individual methods do not need to remember to pass it.