Make errlog optional in stdio_client and update default to None

[sesajad]

I have modified the input type of the errlog parameter in stdio_client from TextIO to TextIO | None, and updated its default value from sys.stderr to None.
Furthermore, I have included a server that generates both logging and stderr prints, along with a client demonstrating how to capture each type of log from the server.

Motivation and Context

Discussed in #1806

How Has This Been Tested?

Testing was conducted by executing examples/snippets/clients/logging_client.py with varying values for errlog.

Breaking Changes

All MCP stdio clients that previously relied on stderr logs being piped to the process stderr will now need to modify their calls:

stdio_client(server_params)

to

import sys
...
stdio_client(server_params, errlog=sys.stderr)

Types of changes

• Bug fix (non-breaking change which fixes an issue)
• New feature (non-breaking change which adds functionality)
• Breaking change (fix or feature that would cause existing functionality to change)
• Documentation update

Checklist

• I have read the MCP Documentation
• My code follows the repository's style guidelines
• New and existing tests pass locally
• I have added appropriate error handling
• I have added or updated documentation as needed

Additional context

None