Definition of a firm

At the Entrepreneur website Per Bylund, author of The Problem of Production: A new theory of the firm (well worth a read if you are into the theory of the firm), has an article on 4 Reasons You Should Start a Firm Instead of Launching a Startup.

Bylund opens the article by saying

Should your next startup be a firm? In conversational usage, the terms are synonyms, so the question might seem odd.

Economically speaking, a firm is a for-profit entity detached from market solutions of the day. The label implies some sort of innovation is happening behind the business’s doors, whether through novel production techniques, new organizational patterns or previously unseen products. (Emphasis added)

The point about the bit in bold is that I'm not sure its a useful definition of a firm.

Consider, for example, a market where you have for-profit entities and not-for-profit entities competing. In the healthcare market you could have for-profit hospitals going up against not-for-profit hospitals. Both groups could be doing the same thing in the same way using the same types of equipment. They may even share some staff in common. So is saying that one is a firm while one isn't really useful?

Or consider a law "firm" which makes profit out of some clients (and then some!) but works pro bono for others. Is the "firm" a firm for some part of the day and not-a-firm for other parts of the day?

Or what of a software company that sells some software but gives away other software. Is it a firm for part of its product line but not for other parts?

In short, I'm not sure for-profit restriction is a useful one.