A public school in America (but not in England) is a state run school which may or may not have free lunches (it's often income related), Private schools are ones you pay to go to and tend to charge for food or order out for food.
It’s not often income-related, it’s always income-related. All public schools offer free/reduced price lunches and even schools in high-income areas will have some small number of students who qualify:
I would have said that but for the email I recently got informing me that all students were getting free lunchs in my son's district. I think it's a covid related state program maybe?
