Churches and taverns had a complicated relationship in Colonial America. As early as 1656, it was a finable offense in Puritan Massachusetts for a town not to have an ordinary.
As you can see by their definitions, the words for a drinking-eating-lodging establishment are mostly interchangeable. (Ordinary became the regional word for a tavern throughout New England.) However, only places called “inns” were reliably somewhere to stay while switching horses or waiting for one’s horse to rest for the next length of travel.