Address review comments and update after docker-in-docker changes

Author: ldionne

.github/workflows/libcxx-build-containers.yaml

@@ -3,8 +3,7 @@
 #
 # The images are pushed to the LLVM package registry at https://github.com/orgs/llvm/packages
 # and tagged appropriately. The selection of which Docker image version is used by the libc++
-# CI nodes at any given point is controlled by the libc++ maintainers using a mechanism that
-# is not publicly visible for safety purposes.
+# CI nodes at any given point is controlled from the workflow files themselves.
 
 name: Build Docker images for libc++ CI
 
@@ -27,25 +26,29 @@ jobs:
     steps:
     - uses: actions/checkout@v4
 
+    - name: Build the Linux builder image and the Android builder image
+      working-directory: libcxx/utils/ci
+      run:
+        - docker compose build actions-builder
+        - docker compose build android-buildkite-builder
+
     - name: Log in to GitHub Container Registry
       uses: docker/login-action@v3
       with:
         registry: ghcr.io
         username: ${{ github.actor }}
         password: ${{ secrets.GITHUB_TOKEN }}
 
-    - name: Build and push the Linux builder image
+    - name: Push the Linux builder image
       working-directory: libcxx/utils/ci
       run:
-        - docker compose build actions-builder
         - docker compose push actions-builder
       env:
         TAG: libcxx-linux-builder:${{ github.sha }}
 
-    - name: Build and push the Android builder image
+    - name: Push the Android builder image
       working-directory: libcxx/utils/ci
       run:
-        - docker compose build android-buildkite-builder
         - docker compose push android-buildkite-builder
       env:
         TAG: libcxx-android-builder:${{ github.sha }}