Fix build-and-deploy-docs workflow (#22966)

We want to publish docs on a successful PR merge to `master`. The
current checks were not triggering on PR merge.

.github/workflows/deploy-docs-site.yml

@@ -9,6 +9,7 @@ on:
 
   pull_request:
     types:
+      - closed
       - opened
       - reopened
       - synchronize
@@ -42,7 +43,7 @@ jobs:
           GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }}
         run: |-
           export SKIP_DEPLOY="yes"
-          if [ "${{github.event_name}}" = 'push' -o "${{github.event_name}}" = 'workflow_dispatch' ]; then
+          if [ "${{github.event_name}}" = 'push' -o "${{github.event_name}}" = 'workflow_dispatch' -o "${{github.event.pull_request.merged}}" = 'true']; then
               export SKIP_DEPLOY="no"
           fi