refactor: reorganize message handling for better type safety and clarity (#239)

* refactor: improve typing with memory stream type aliases

Move memory stream type definitions to models.py and use them throughout
the codebase for better type safety and maintainability.

GitHub-Issue:#201

* refactor: move streams to ParsedMessage

* refactor: update test files to use ParsedMessage

Updates test files to work with the ParsedMessage stream type aliases
and fixes a line length issue in test_201_client_hangs_on_logging.py.

Github-Issue:#201

* refactor: rename ParsedMessage to MessageFrame for clarity

🤖 Generated with [Claude Code](https://claude.ai/code)
Co-Authored-By: Claude <noreply@anthropic.com>

* refactor: move MessageFrame class to types.py for better code organization

🤖 Generated with [Claude Code](https://claude.ai/code)
Co-Authored-By: Claude <noreply@anthropic.com>

* fix pyright

* refactor: update websocket client to use MessageFrame

Modified the websocket client to work with the new MessageFrame type,
preserving raw message text and properly extracting the root JSON-RPC
message when sending.

Github-Issue:#204

* fix: use NoneType instead of None for type parameters in MessageFrame

🤖 Generated with [Claude Code](https://claude.ai/code)

* refactor: rename root to message

src/mcp/server/sse.py

@@ -38,7 +38,6 @@ from urllib.parse import quote
 from uuid import UUID, uuid4
 
 import anyio
-from anyio.streams.memory import MemoryObjectReceiveStream, MemoryObjectSendStream
 from pydantic import ValidationError
 from sse_starlette import EventSourceResponse
 from starlette.requests import Request
@@ -46,6 +45,13 @@ from starlette.responses import Response
 from starlette.types import Receive, Scope, Send
 
 import mcp.types as types
+from mcp.shared.session import (
+    ReadStream,
+    ReadStreamWriter,
+    WriteStream,
+    WriteStreamReader,
+)
+from mcp.types import MessageFrame
 
 logger = logging.getLogger(__name__)
 
@@ -63,9 +69,7 @@ class SseServerTransport:
     """
 
     _endpoint: str
-    _read_stream_writers: dict[
-        UUID, MemoryObjectSendStream[types.JSONRPCMessage | Exception]
-    ]
+    _read_stream_writers: dict[UUID, ReadStreamWriter]
 
     def __init__(self, endpoint: str) -> None:
         """
@@ -85,11 +89,11 @@ class SseServerTransport:
             raise ValueError("connect_sse can only handle HTTP requests")
 
         logger.debug("Setting up SSE connection")
-        read_stream: MemoryObjectReceiveStream[types.JSONRPCMessage | Exception]
-        read_stream_writer: MemoryObjectSendStream[types.JSONRPCMessage | Exception]
+        read_stream: ReadStream
+        read_stream_writer: ReadStreamWriter
 
-        write_stream: MemoryObjectSendStream[types.JSONRPCMessage]
-        write_stream_reader: MemoryObjectReceiveStream[types.JSONRPCMessage]
+        write_stream: WriteStream
+        write_stream_reader: WriteStreamReader
 
         read_stream_writer, read_stream = anyio.create_memory_object_stream(0)
         write_stream, write_stream_reader = anyio.create_memory_object_stream(0)
@@ -172,4 +176,4 @@ class SseServerTransport:
         logger.debug(f"Sending message to writer: {message}")
         response = Response("Accepted", status_code=202)
         await response(scope, receive, send)
-        await writer.send(message)
+        await writer.send(MessageFrame(message=message, raw=request))