When One Says" Publish a Book" does this mean Pay For It?

I was asked this question: Now with all of these Print on Demand publishers, and the ones who charge you a lot of money, a lot of money sometimes ( read vanity presses) then when someone says i published a book, does this mean pay for it?

Well to me no. If you publish a book you should only pay for what you get, which is books, and that is really paying for the cost, the profit is yours ( given the style of question I am assuming self-publishing)

Still, you might even go to a traditional publisher and not have to pay for your publishing costs. There is a hidden price, your time and energy to sell the books and make back your advance and then some.

So when one says "publish a book, does this mean pay of it? the answer is yes.