That's a fairly succinct definition.

There are non-profits that make millions of dollars in positive cash flow. All that term means (At least in the US) is that it doesn't ever pay dividends to shareholders.

I'll be intentionally vague because I don't want to stray too far afield but there are some large organization that make a lot of money but are classified as non-profit. They can pay excess revenue as bonuses to directors and executives.