May still be corporate, because corporations are a way of structuring a business, not an indication of how large or how many people are involved with the business. I expect most family farms are also corporations or even multiple corporations.

No. Most family farms, at least in my part of the states, are not corporations. They are small businesses in that they have some tax exemptions for being farms, but they are not incorporated.

Only the 'slick' farmers (you know, them city folks) or large businesses that are 'family farms' in name only are incorporated.

In fact, I know that several of my neighbours have a corporation setup for each generation of the family farm. These aren't slick city folks or even large businesses. They're every day family farmers that farm the exact same way I do (my farm isn't incorporated). But, like you suggest, maybe it is regional thing.

A 1668 acre farm around here is a small farm, and is probably not incorporated. But certainly many are. But as you said, incorporation does not make it less of a family farm.