fix(rust): Add cargo publish step to release workflow

[konard]

Summary

This PR fixes the missing crates.io publishing step in the Rust release workflow.

Problem

The release workflow correctly detected that version 0.4.0 was not published to crates.io:

Crate: browser-commander, Version: 0.4.0, Published on crates.io: false
No changelog fragments but v0.4.0 not yet published to crates.io

However, the workflow only:

1. Built the release binary (cargo build --release)
2. Created a GitHub release

The actual cargo publish step was missing, so the package was never published to crates.io.

Solution

Added the missing Publish to Crates.io step to both auto-release and manual-release jobs in the workflow, following best practices from the template repository.

Changes

• rust/scripts/publish-crate.mjs: New script to handle crates.io publishing

  • Supports both single-language and multi-language repositories
  • Handles "already exists" case gracefully
  • Uses CARGO_REGISTRY_TOKEN or CARGO_TOKEN environment variable

• rust/scripts/rust-paths.mjs: Helper module for multi-language repo support

  • Auto-detects repository structure (root vs rust/ subfolder)
  • Provides utilities for path resolution

• .github/workflows/rust.yml: Added Publish to Crates.io step

  • Added after Build release, before Create GitHub Release
  • Added to both auto-release and manual-release jobs

• docs/case-studies/issue-31/: Case study documentation

  • Timeline and root cause analysis
  • Reference to CI logs and related issues

Repository Secret Requirements

The repository needs CARGO_REGISTRY_TOKEN secret configured for crates.io authentication.

Test Plan

• YAML syntax validated
• JavaScript syntax validated for new scripts
• Workflow structure verified against template repository
• Integration test (requires CI run after merge)

Related Issues

• Rust release jobs skipped #27: Rust release jobs skipped
• Release failed, because version check got false positive #29: Release failed, because version check got false positive
• I see not actual publishing to crates.io #31: I see not actual publishing to crates.io (this PR)

References

• Template repository: https://github.com/link-foundation/rust-ai-driven-development-pipeline-template
• CI Run showing the problem: https://github.com/link-foundation/browser-commander/actions/runs/21103777967

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Fixes #31

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[konard]

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