ci: deploy fails because of missing public/ dir

Problem:
Since moving to `peaceiris/actions-gh-pages` #401
there is a new problem:

    /home/runner/actions_github_pages_1764026989086
      cp: no such file or directory: /home/runner/work/neovim.github.io/neovim.github.io/public/*
      cp: no such file or directory: /home/runner/work/neovim.github.io/neovim.github.io/public/.*

The old way magically worked because `actions/upload-pages-artifact` is
internally handled by `actions/deploy-pages`. There is no complementary
`actions/download-pages-artifact` so we have to do some other approach.

Solution:
Do the build in the deploy step, avoid uploading/downloading artifacts.

Author: justinmk

.github/actions/build-site/action.yaml

@@ -0,0 +1,23 @@
+name: "Build Hugo Site"
+description: "Shared Hugo build logic"
+
+runs:
+  using: "composite"
+  steps:
+    - name: Install Hugo CLI
+      shell: bash
+      env:
+        HUGO_VERSION: "0.152.2"
+      run: |
+        wget -O ${{ runner.temp }}/hugo.deb \
+          https://github.com/gohugoio/hugo/releases/download/v${HUGO_VERSION}/hugo_extended_${HUGO_VERSION}_linux-amd64.deb \
+        && sudo dpkg -i ${{ runner.temp }}/hugo.deb
+    - name: Build with Hugo
+      shell: bash
+      env:
+        # For backward compatibility with Hugo modules
+        HUGO_ENVIRONMENT: production
+        HUGO_ENV: production
+      run: |
+        hugo --gc --minify --baseURL "$BASE_URL"
+

.github/workflows/build.yaml

@@ -0,0 +1,21 @@
+# Builds but does not deploy, for pull request validation.
+
+name: build
+
+on:
+  # Runs on pull requests
+  pull_request:
+
+jobs:
+  build:
+    runs-on: ubuntu-latest
+    steps:
+      - name: Checkout
+        uses: actions/checkout@v6
+        with:
+          submodules: recursive
+          fetch-depth: 0
+      - name: Build site
+        uses: ./.github/actions/build-site
+        env:
+          BASE_URL: "https://example.com/"

.github/workflows/hugo.yaml renamed to .github/workflows/deploy.yaml

@@ -1,3 +1,6 @@
+# Builds and deploys the site to GitHub Pages.
+# Does NOT run on pull requests.
+
 name: build-and-deploy
 
 on:
@@ -6,9 +9,6 @@ on:
     branches:
       - master
 
-  # Runs on pull requests
-  pull_request:
-
   # Allows you to run this workflow manually from the Actions tab
   workflow_dispatch:
 
@@ -24,56 +24,28 @@ concurrency:
   group: "pages"
   cancel-in-progress: false
 
-# Default to bash
-defaults:
-  run:
-    shell: bash
-
 jobs:
-  # Build job
-  build:
+  # Deployment job
+  build-and-deploy:
     runs-on: ubuntu-latest
-    env:
-      HUGO_VERSION: 0.152.2
+    permissions:
+      contents: write
+    environment:
+      name: github-pages
+      url: ${{ steps.deployment.outputs.page_url }}
     steps:
-      - name: Install Hugo CLI
-        run: |
-          wget -O ${{ runner.temp }}/hugo.deb https://github.com/gohugoio/hugo/releases/download/v${HUGO_VERSION}/hugo_extended_${HUGO_VERSION}_linux-amd64.deb \
-          && sudo dpkg -i ${{ runner.temp }}/hugo.deb
       - name: Checkout
         uses: actions/checkout@v6
         with:
           submodules: recursive
           fetch-depth: 0
-      - name: Build with Hugo
+      - name: Build site
+        uses: ./.github/actions/build-site
         env:
-          # For maximum backward compatibility with Hugo modules
-          HUGO_ENVIRONMENT: production
-          HUGO_ENV: production
-        run: |
-          hugo \
-            --gc \
-            --minify \
-            --baseURL "${{ steps.pages.outputs.base_url }}/"
-      - name: Upload artifact
-        uses: actions/upload-pages-artifact@v4
-        with:
-          path: ./public
-
-  # Deployment job
-  deploy:
-    if: github.event_name == 'push' && github.ref == 'refs/heads/master'
-    environment:
-      name: github-pages
-      url: ${{ steps.deployment.outputs.page_url }}
-    runs-on: ubuntu-latest
-    permissions:
-      contents: write
-    needs: build
-    steps:
-      - name: Deploy to GitHub Pages
+          BASE_URL: "${{ steps.pages.outputs.base_url }}/"
+      - name: Deploy
         id: deployment
         uses: peaceiris/actions-gh-pages@v4
         with:
-          github_token: ${{ secrets.GITHUB_TOKEN }}
           publish_dir: ./public
+          github_token: ${{ secrets.GITHUB_TOKEN }}