If you leave out the storage class in your manifest and if your cluster admin has set up a default storage class (which is not identified by the name standard but by an annotation on the storage class ) it will work without any explicit storage class setup.
Our Quickstart example does not have an explicit volumeClaimTemplate but the documentation you linked to Volume claim templates | Elastic Cloud on Kubernetes [2.2] | Elastic could indeed be improved as it does not explain where the standard storage class comes from. I will open an issue to get this updated.