Update _client_info on the ClientSession upon initialization with serverInfo #1018

            transport = await self.exit_stack.enter_async_context(
                streamablehttp_client(server_url, headers=headers)
            )
            self.read, self.write, _ = transport
            self.session = await self.exit_stack.enter_async_context(
                ClientSession(self.read, self.write)
            )

            server = await self.session.initialize()
            self.name = server.serverInfo.name  # <====

class ClientSession(
    BaseSession[
        types.ClientRequest,
        types.ClientNotification,
        types.ClientResult,
        types.ServerRequest,
        types.ServerNotification,
    ]
):
    def __init__(
        self,
        read_stream: MemoryObjectReceiveStream[SessionMessage | Exception],
        write_stream: MemoryObjectSendStream[SessionMessage],
        # ...
        client_info: types.Implementation | None = None,
    ) -> None:
        super().__init__(...)
        self._client_info = client_info or DEFAULT_CLIENT_INFO # <<<<<<
       # ...

    async def initialize(self) -> types.InitializeResult:
        # ...
        result = await self.send_request(
            types.ClientRequest(
                types.InitializeRequest(
                    method="initialize",
                    params=types.InitializeRequestParams(
                        protocolVersion=types.LATEST_PROTOCOL_VERSION,
                        capabilities=types.ClientCapabilities(
                            sampling=sampling,
                            experimental=None,
                            roots=roots,
                        ),
                        clientInfo=self._client_info,
                    ),
                )
            ),
            types.InitializeResult,
        )
        # ...
        return result