docs: use official GitHub Action to deploy to GitHub Pages

[vlad-nestorov]

Pre-flight checklist

• I have read the Contributing Guidelines on pull requests.
• If this is a code change: I have written unit tests and/or added dogfooding pages to fully verify the new behavior.
• If this is a new API or substantial change: the PR has an accompanying issue (closes #0000) and the maintainers have approved on my working plan.

Motivation

Update Github Actions deployment instructions to use the more Github way of doing things, and remove dependency on third party actions.

Test Plan

See this repo for an example successful run of the new instructions: https://github.com/vlad-nestorov/vlad-nestorov.github.io/actions/runs/8224051582/workflow

Test links

https://deploy-preview-9937--docusaurus-2.netlify.app/docs/deployment#triggering-deployment-with-github-actions

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[github-actions]

⚡️ Lighthouse report for the deploy preview of this PR

URL	Performance	Accessibility	Best Practices	SEO	PWA	Report
/	🟠 53	🟢 98	🟢 96	🟢 100	🟠 88	Report
/docs/installation	🟠 54	🟢 96	🟢 100	🟢 100	🟠 88	Report
/docs/category/getting-started	🟠 76	🟢 100	🟢 100	🟢 90	🟠 88	Report
/blog	🟠 69	🟢 100	🟢 100	🟢 90	🟠 88	Report
/blog/preparing-your-site-for-docusaurus-v3	🟠 62	🟢 96	🟢 100	🟢 100	🟠 88	Report
/blog/tags/release	🟠 69	🟢 100	🟢 100	🟠 80	🟠 88	Report
/blog/tags	🟠 74	🟢 100	🟢 100	🟢 90	🟠 88	Report

[facebook-github-bot]

Thank you for signing our Contributor License Agreement. We can now accept your code for this (and any) Meta Open Source project. Thanks!