[theamodhshetty]

Clarify when ServiceExt::serve() returns and how to run the counter stdio example with a real MCP client.

Motivation and Context

Refs #1143. Starting the counter executable directly in a terminal can leave serve().await waiting for peer input, which looks like a stalled server. The API docs, README and example now explain protocol startup, background message handling, waiting(), startup cancellation and Tokio stdin's blocking-read shutdown limitation. The example also gets a working Inspector command and a corrected stderr logging comment.

This addresses the issue's documentation request; lifecycle and shutdown behavior are unchanged.

How Has This Been Tested?

On macOS arm64 with Rust 1.96.1:

• All 22 existing server-initialization tests passed.
• Built servers_counter_stdio and ran the documented command through Inspector CLI 2.12.0: initialization succeeded, tools/list returned eight tools, and the service stopped after input-stream closure.
• Built rmcp API docs with broken intra-doc links denied and the relevant transport features enabled.
• Nightly workspace formatting and git diff --check passed.

Breaking Changes

None; documentation and comments only.

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

No runtime error-handling changes are involved. Initially submitted as a draft under the repository's one-open-ready-PR limit. #1386 has now merged, so this documentation change is ready for maintainer review. Hosted CI, conformance, CodeQL and Scorecard now pass on a0fdcad; maintainer review remains pending. Developed with Codex assistance; validation was run locally.

[github-actions]

@DaleSeo (assignment failed) is on triage rotation this week and is the reviewer for this one.

Labels come from the PR labeler. Please review the change.

_{Automated triage. Labels are a first pass and the assigned maintainer has the final say.}