I announced through a loudspeaker that my son had embarrassing habits at home
Why would you want to do something like that to him?
He told me there is nothing wrong with that. What can I say?
Maybe you misread what he was trying to say. I don't feel like your son would be very receptive to being publicly shamed.
I'm going to punish him for setting a bad example for other people.
I don't think punishing him in this instance would be beneficial to either party. Like I said there seems to be some form of miscommunication here.