update(Github Action): Remove old GitHub action references and replace with the new CLI action

[BartoszBlizniak]

As we will be deprecating the old GHA, the documentation needs to be updated and re-pointed to the new action.

[vercel]

The latest updates on your projects. Learn more about Vercel for GitHub.

Project	Deployment	Preview	Comments	Updated (UTC)
cloudsmith-docs	Ready	Preview	Comment	Oct 23, 2025 11:35am

[Copilot]

Pull Request Overview

This PR updates the GitHub Actions integration documentation to reflect the transition from the deprecated cloudsmith-io/action to the new cloudsmith-io/cloudsmith-cli-action. The documentation now emphasizes OIDC authentication as the recommended approach and provides comprehensive examples for various authentication methods.

Key Changes:

• Replaced references to the old GitHub Action with the new CLI-based action
• Added detailed OIDC authentication documentation with code examples
• Included comprehensive workflow examples showing real-world usage patterns

---

_{Tip: Customize your code reviews with copilot-instructions.md. Create the file or learn how to get started.}