Consider the category of presheaves C^ on a small category, C.

Plainly, a finite colimit F of representables presheaves is finitely presentable, in the usual sense: C^(F, -) preserves filtered colimits. But the converse is also true and seemingly well known: every finitely presentable presheaf is a finite colimit of representables.

Is this proved somewhere?

I think this is obvious from the fact that the theory of presheaves over C is many sorted, essentially algebraic (a finite limit theory), with a sort for each object of C and a unary operator for each morphism. Then finitely presentable in the categorical sense is the same as finitely presentable in the algebraic sense, which is equivalent to being a finite colimit of free cyclic (i.e. one generator) algebras. In the case of presheaves, Yoneda's lemma says precisely that the free algebra on a generator of sort X (object of C) is the representable presheaf for X. I have exploited some of these facts in a paper with my PhD student Gillian Hill, "Presheaves as configured specifications". It develops a language for specifying systems by components with sharing. Steve Vickers.