feat(vertexai): organize example page and functions (#17008)

* organize example page and functions

* fix analyzer

* review feedback update

Author: cynthiajoan

packages/firebase_vertexai/firebase_vertexai/example/.gitignore

@@ -5,9 +5,11 @@
 *.swp
 .DS_Store
 .atom/
+.build/
 .buildlog/
 .history
 .svn/
+.swiftpm/
 migrate_working_dir/
 
 # IntelliJ related