Align stdio shutdown sequence with spec

[davenpi]

Align the stdio shutdown sequence with the MCP spec: close the server’s input, wait for exit, send SIGTERM if needed, and SIGKILL as a last resort.

Motivation and Context

The MCP spec recommends a careful shutdown sequence for stdio transport to avoid leaving orphaned or stuck server processes. This change follows those steps, making shutdowns more reliable and predictable.

How Has This Been Tested?

All tests pass (except one unrelated skipped test).

Breaking Changes

None

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

[davenpi]

Hey @ihrpr! Do you have a sec to review this PR? It makes the SDK follow the stdio shutdown procedure to the letter!

[ihrpr]

related PR: #555

[ihrpr]

Thank you @davenpi for working on this.

There are a lot of issues reported around the shutdown for stdio, if would be great to have additional tests and verify on different platforms that the solution works.