docs: use official GitHub Action to deploy to GitHub Pages (#9937)

Co-authored-by: ozakione <29860391+OzakIOne@users.noreply.github.com>

website/docs/deployment.mdx

@@ -348,17 +348,17 @@ Alternatively, you can use SSH (`USE_SSH=true`) to log in.
 
 [GitHub Actions](https://help.github.com/en/actions) allow you to automate, customize, and execute your software development workflows right in your repository.
 
-The workflow examples below assume your website source resides in the `main` branch of your repository (the _source branch_ is `main`), and your [publishing source](https://help.github.com/en/github/working-with-github-pages/configuring-a-publishing-source-for-your-github-pages-site) is configured for the `gh-pages` branch (the _deployment branch_ is `gh-pages`).
+The workflow examples below assume your website source resides in the `main` branch of your repository (the _source branch_ is `main`), and your [publishing source](https://help.github.com/en/github/working-with-github-pages/configuring-a-publishing-source-for-your-github-pages-site) is configured for [publishing with a custom GitHub Actions Workflow](https://docs.github.com/en/pages/getting-started-with-github-pages/configuring-a-publishing-source-for-your-github-pages-site#publishing-with-a-custom-github-actions-workflow).
 
 Our goal is that:
 
 1. When a new pull request is made to `main`, there's an action that ensures the site builds successfully, without actually deploying. This job will be called `test-deploy`.
-2. When a pull request is merged to the `main` branch or someone pushes to the `main` branch directly, it will be built and deployed to the `gh-pages` branch. After that, the new build output will be served on the GitHub Pages site. This job will be called `deploy`.
+2. When a pull request is merged to the `main` branch or someone pushes to the `main` branch directly, it will be built and deployed to GitHub Pages. This job will be called `deploy`.
 
 Here are two approaches to deploying your docs with GitHub Actions. Based on the location of your deployment branch (`gh-pages`), choose the relevant tab below:
 
 - Source repo and deployment repo are the **same** repository.
-- The deployment repo is a **remote** repository, different from the source.
+- The deployment repo is a **remote** repository, different from the source. Instructions for this scenario assume [publishing source](https://help.github.com/en/github/working-with-github-pages/configuring-a-publishing-source-for-your-github-pages-site) is the `gh-pages` branch.
 
 ```mdx-code-block
 <Tabs>
@@ -367,8 +367,6 @@ Here are two approaches to deploying your docs with GitHub Actions. Based on the
 
 While you can have both jobs defined in the same workflow file, the original `deploy` workflow will always be listed as skipped in the PR check suite status, which is not indicative of the actual status and provides no value to the review process. We therefore propose to manage them as separate workflows instead.
 
-We will use a popular third-party deployment action: [peaceiris/actions-gh-pages](https://github.com/peaceiris/actions-gh-pages#%EF%B8%8F-docusaurus).
-
 <details>
 <summary>GitHub action files</summary>
 
@@ -392,12 +390,9 @@ on:
     # Review gh actions docs if you want to further define triggers, paths, etc
     # https://docs.github.com/en/actions/using-workflows/workflow-syntax-for-github-actions#on
 
-permissions:
-  contents: write
-
 jobs:
-  deploy:
-    name: Deploy to GitHub Pages
+  build:
+    name: Build Docusaurus
     runs-on: ubuntu-latest
     steps:
       - uses: actions/checkout@v4
@@ -413,21 +408,30 @@ jobs:
       - name: Build website
         run: yarn build
 
-      # Popular action to deploy to GitHub Pages:
-      # Docs: https://github.com/peaceiris/actions-gh-pages#%EF%B8%8F-docusaurus
-      - name: Deploy to GitHub Pages
-        uses: peaceiris/actions-gh-pages@v3
+      - name: Upload Build Artifact
+        uses: actions/upload-pages-artifact@v3
         with:
-          github_token: ${{ secrets.GITHUB_TOKEN }}
-          # Build output to publish to the `gh-pages` branch:
-          publish_dir: ./build
-          # The following lines assign commit authorship to the official
-          # GH-Actions bot for deploys to `gh-pages` branch:
-          # https://github.com/actions/checkout/issues/13#issuecomment-724415212
-          # The GH actions bot is used by default if you didn't specify the two fields.
-          # You can swap them out with your own user credentials.
-          user_name: github-actions[bot]
-          user_email: 41898282+github-actions[bot]@users.noreply.github.com
+          path: build
+
+  deploy:
+    name: Deploy to GitHub Pages
+    needs: build
+
+    # Grant GITHUB_TOKEN the permissions required to make a Pages deployment
+    permissions:
+      pages: write # to deploy to Pages
+      id-token: write # to verify the deployment originates from an appropriate source
+
+    # Deploy to the github-pages environment
+    environment:
+      name: github-pages
+      url: ${{ steps.deployment.outputs.page_url }}
+
+    runs-on: ubuntu-latest
+    steps:
+      - name: Deploy to GitHub Pages
+        id: deployment
+        uses: actions/deploy-pages@v4
 ```
 
 ```yml title=".github/workflows/test-deploy.yml"