Can and ex-employer tell people that you was terminated?

Any time you leave your employer for any reason, that's a termination. If you quit, it's a voluntary termination. If your employer lets you go, it's an involuntary termination.

Yes, your employer may tell others that you were terminated, and they can tell them whether you were voluntarily or involuntarily terminated, and they can tell them WHY you were terminated.

Is this true in all states?

Because here in New York, I was always told that an employer is not at liberty to share with other employees why another employee leaves the company - whether it's voluntary/quitting or being fired/laid off.

There are no laws that direct whether employers can say why an employee left the company. The only laws that remotely touch on this are those related to slander/libel in that an employer cannot knowingly provide false and defamatory information.

Some employers may choose to have a policy that prohibits such discussions, but that's all it is -- a policy.

Then, generally speaking, somebody told you incorrect information.

Yes, that is true in all states.

It is possible for a company to have a policy not to disclose such information. No law in any state prohibits it.