Are businesses included in economic institutions? 1st definition from businessdictionary.com:
Network of commercial organizations (such as manufacturers, producers, wholesalers, retailers, and buyers) who generate, distribute, and purchase goods and services. See also financial institutions.

2nd definition from econlib.org:
1. Specific agencies or foundations, both government and private, devoted to collecting or studying economic data, or commissioned with the job of supplying a good or service that is important to the economy of a country. The Internal Revenue Service (the IRS—the government tax-collection agency), the U.S. Federal Reserve (the government producer of money), the National Bureau of Economic Research (a private research agency) are all examples of economic institutions.
2. Well-established arrangements and structures that are part of the culture or society, e.g., competitive markets, the banking system, kids' allowances, customary tipping, and a system of property rights are examples of economic institutions.

Which one is correct? Also, on wikipedia it says businesses are a part of institutions. https://en.wikipedia.org/wiki/Institution
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.