Use BABEL_BOT token to push types updates to babel/website

.github/workflows/release.yml

@@ -280,8 +280,6 @@ jobs:
 
   types-docs-updates:
     name: Update Babel types docs
-    permissions:
-      contents: write # for Git to git push
     runs-on: ubuntu-latest
     needs:
       - check-release-type
@@ -297,6 +295,8 @@ jobs:
       - uses: actions/checkout@v4
         with:
           repository: babel/website
+          fetch-depth: 0 # Otherwise we cannot push
+          persist-credentials: false # So that we can push with BOT_TOKEN, otherwise it doesn't trigger CI
       - name: Download babel-types docs
         uses: actions/download-artifact@v3
         with:
@@ -309,7 +309,7 @@ jobs:
           git config user.email "babel-bot@users.noreply.github.com"
           git checkout -b update-types-docs
           git commit -am "docs: update babel types"
-          git push --force origin update-types-docs
+          git push --force "https://babel-bot:${{ secrets.BOT_TOKEN }}@github.com/babel/website.git" update-types-docs
       - name: Create Pull Request
         uses: babel/actions/create-pull-request@v2
         env: