Update docs-elastic-staging-publish.yml

Done as part of our serverless move (https://github.com/elastic/docs-projects/issues/178)

- Copies the public docs builder for a public repo (https://github.com/elastic/workflows/tree/main?tab=readme-ov-file#public-repos)
- Updates paths to point at /docs/serverless

.github/workflows/docs-elastic-staging-publish.yml

@@ -1,25 +1,29 @@
-name: Staging Docs
+name: Elastic docs
 
 on:
   pull_request_target:
+    # The paths property can be omitted entirely if the repo is mainly used for docs. Leaving it in can result in builds that 
+    # have branch protection checks in place lose the ability to merge because the workflow is not starting. If this property 
+    # is included, please ensure that branch protection checks are disabled for the repo. 
     paths:
-      - '**.mdx'
-      - '**.docnav.json'
-      - '**.docapi.json'
-      - '**.devdocs.json'
-      - '**.jpg'
-      - '**.jpeg'
-      - '**.png'
-      - '**.svg'
-      - '**.gif'
-    types: [opened, closed, synchronize]
+      # Preface with your docs dir if you need further specificity (optional)
+      - 'docs/serverless/**.mdx'
+      - 'docs/serverless/**.docnav.json'
+      - 'docs/serverless/**.docapi.json'
+      - 'docs/serverless/**.devdocs.json'
+      - 'docs/serverless/**.jpg'
+      - 'docs/serverless/**.jpeg'
+      - 'docs/serverless/**.svg'
+      - 'docs/serverless/**.png'
+      - 'docs/serverless/**.gif'
+    types: [closed, opened, synchronize, labeled]
 
 jobs:
   publish:
-    name: Vercel Build Check
+    if: contains(github.event.pull_request.labels.*.name, 'ci:doc-build')
     uses: elastic/workflows/.github/workflows/docs-elastic-co-publish.yml@main
     secrets:
-      VERCEL_GITHUB_TOKEN: ${{ secrets.VERCEL_GITHUB_TOKEN }}
-      VERCEL_TOKEN: ${{ secrets.VERCEL_TOKEN }}
-      VERCEL_ORG_ID: ${{ secrets.VERCEL_ORG_ID }}
+      VERCEL_GITHUB_TOKEN: ${{ secrets.VERCEL_GITHUB_TOKEN_PUBLIC }}
+      VERCEL_TOKEN: ${{ secrets.VERCEL_TOKEN_PUBLIC }}
+      VERCEL_ORG_ID: ${{ secrets.VERCEL_ORG_ID_PUBLIC }}
       VERCEL_PROJECT_ID_DOCS_CO: ${{ secrets.VERCEL_PROJECT_ID_DOCS_CO }}