OK, dueling definitions it is:

"An organization or economic system where goods and services are exchanged for one another or for money.
Every business requires some form of investment and enough customers to whom its output can be sold on a consistent basis in order to make a profit.
Businesses can be privately owned, not-for-profit or state-owned. An example of a corporate business is PepsiCo, while a mom-and-pop catering business is a private enterprise."
http://www.businessdictionary.com/definition/business.html"

The essential difference is whether the "objective of making a profit" is essential to a business being a business.  I say yes. 

More precisely, the issue is to grasp that some, but not all, non-profit entities may have the "objective of profit" in some of their activities, but still not be profit-driven.

I do concede that fully private universities can be construed as businesses in this sense, but that the Church and publicly funded universities do not FULLY fit.