Theophilus

They were originally a not-for-profit but now are a private, for profit company.

Yes they are a private limited company because they aim to make a profit

a private company, is a company or group of companies owned by a single person or a group of people (share holders), they collect its profit based on an understanding they have.

a public company is usually a listed company or a government owned company, where its profit are usually collected by the government.

Companies are responsible to their shareholders (or owners in a private company) for making a profit and to governments for obeying the laws.