Configure hosted stdio servers to log to stderr

[halter73]

No description provided.

[stephentoub]

Have you tested how this behaves with various tools as clients?

What's the experience going to be like with our client, that does this:

csharp-sdk/src/ModelContextProtocol/Protocol/Transport/StdioClientTransport.cs

Line 88 in 96fa96c

process.ErrorDataReceived += (sender, args) => logger.TransportError(endpointName, args.Data ?? "(no data)");

?

[halter73]

It works well with the server inspector. I think we should update ErrorDataReceived event to emit a new info-level log and give it a new message specific to to our client receiving stderr output. I can do that in this PR.