Are businesses included in economic institutions?
Unfortunately, economists don't have a standard, universally-accepted definition of what an 'institution' is.

Hence, depending on what you're trying to study and whom you ask, businesses may or may not be an institution. So, in a way, both definitions are correct.

If you're interested in different conceptual approaches to institutions, Avner Greif has written an interesting survey on it, with plenty of references that might be worth looking at.