I wasn't HR. It wasn't HR calling HR to verify employment. It was a former customer calling for a reference. So yes it can happen that if that OP used the company as a reference, the former manager could take the call and say, "He quit without 2 weeks notice." It isn't standard as your own link confirmed. But that's not hr verifying employment.
"Terminated with cause”
This wasn't terminated with cause, fired or anything like that. The post is about an employee who quit. They are allowed to quit.