You have to be in at least 72 hours voluntary. Unless, you show a danger, you can leave on your own after that time.

Harding says to McMurphy that he is not committed, but voluntary and that he can go home anytime he want.
So this means that Harding is still a patient at the ward, but he is allowed to go home for visits? Or does he have the choice to make himself discharged from the hospital altogether
Not quite that easy. A 30 day notice needs to be written by the patient for discharge, but if the staff feels that after you have voluntarily admitted yourself that you are a risk, that notice doesn't have to be honored.
And once you are admitted as an in-patient, you cannot go home for visits at first, whether it's voluntary or involuntary

I've stayed at mental wards myself
Were you voluntary?
Each time I've been in, I was detained for a couple of months. I was voluntary around 1-2 weeks prior to being discharged.

so you mis-quote? Not the way it works.

imdb has a 100-character limit in the signature.
Scariest words in English: Were from the federal government and were here to help. R. Reagan

btw- You've misquoted Ronald Reagan. The actual quote is;
"The nine most terrifying words in the English language are: I'm from the government and I'm here to help."

I've stayed at mental wards myself
Were you voluntary?
Scariest words in English: Were from the federal government and were here to help. R. Reagan

Harding is there as a patient under his own volition. He can check himself out of the hospital at his discretion. Murphy has been committed to the hospital by the court. If he leaves or refuses treatment he goes to jail.

It means he can leave anytime they want. It's likely the rules in the US are similar to Australian Mental hospitals.