From: Boerger <Reinhard.Boerger@FernUni-Hagen.de> the simplest characterization of FinSet known to me is as the free category with initial object and binary coproducts on one object.
Intuitively initiality seems like a bigger hammer than finiteness. This is confirmed by the observation that, although both initiality and finiteness properly extend first order logic as a specification methodology, initiality *guarantees* uniqueness up to whatever. With the assumption only of finiteness of objects, uniqueness is not so easily obtained (cf linear orders without endpoints, an alpha-null-categorical theory). It's like driving to work vs. riding a bicycle. Driving is faster and more convenient, the bicycle shifts more of the responsibility to the rider.
In the usual world existence of these types of coproducts is equivalent to finite coproducts, but restriction to nullary and binary ones avoids the need for an a priori notion of finiteness.
A nice point and I did consider writing "with binary sums and an initial object" in place of "with finite sums" at the time. Given the brevity of the latter it might be preferable to consider it defined as the former by default, indicating any exceptions explicitly.
Somehow this reminds me of Kuratowski's definition of finiteness.
I thought Kuratowski's idea was that when a shepherd verifies that he has only finitely many sheep by completing the process of branding them all, it is not necessary when branding a given sheep to first check whether it has already been branded. All that matters (to both the shepherd and the sheep) is that the branding eventually stop. Here's a definition for traditionalist physicists etc. who believe that God created the continuum while the natural numbers are the work of mankind. ** A set of points is finite just when its members can be positioned with equal ** nonzero spacing in a straight line across an A4 sheet of paper. This is the appropriate converse to the Archimedean axiom, violated only for those physicists who view the continuum as including infinitesimals. Pace Joe Shipman, any physicist demonstrating this deserves a Nobel prize. Somehow this reminds me of how many angels can fit on the head of a pin. Vaughan