Employment verification can absolutely include a description of the separation, eg "resignation with notice," "resignation without notice," or "terminated for cause." Lots of people saying this can't be said, but no one has cited any source because it's false
Your link isn't relevant because it is about being fired, not quitting.
But from your link:
"Many organizations have policies that limit their staff to providing only dates of employment and job titles when inquiries are made about past employees. Others may be more willing to share information with prospective employers."
So your statement only applies to your situation as your link confirms.
I hired people too. In only one situation where any ex employee was caught stealing from a client did I ever share anything negative. And that was only because the company who called was also a former customer.
I like how you end your post with an example of you doing the thing you're saying companies don't do lol
Of course this varies by organization and policy, but it's a real thing that many companies do. "Terminated with cause" aka fired is just one of several separation categories companies may share
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.
I don't know what you're arguing against here, my entire point is that there are some companies that communicate separation status by policy, whether that's "terminated for cause," "resigned without notice," or "resigned with notice," and that "resigned without notice" can negatively impact job opportunities at some companies.
If someone tries to verify employment at my previous job, they will be directed to https://theworknumber.com/, which a metric fuckton of businesses use. When they access the information on that site, they will find my company's HR department has uploaded the name, dates of employment, and reason for separation. I don't understand why you think I'm somehow a proponent or advocate of this, I'm literally just communicating that it exists and is real
Regarding not sharing your previous employer, you're welcome to choose to lie on your resume, but I'm not going to