Can someone complete and correct this definition of a finitary functor and perhaps give a link to the definition?
$F:\text{Set}\to \text{Set}$ is finitary iff for every $X$ and $x\in X$ there is a finite $Y$ and $i:Y\to X$ with $Fx\in Fi[FY]$ .
The situation is that I've seen this at the blackboard in a seminar on category theory but haven't caught the details of the defintion. The equivalent definition of a finitary functor should be that it preserves directed colimits, but I do not see the equivalence of these two definitions (this and the one almost correct above) either.
I think you have some typos in your equivalence, and I guess that you wanted to asking for the equivalence:
$F\colon \mathsf{Set}\to\mathsf{Set}$ is finitary iff for every $X$ and $x\in FX$ there is a finite $Y$ and $i\colon Y\to X$ with $x\in Fi[FY]$
(By brackets I mean the direct image notation $Fi[FY] \subseteq FX$)
Regarding the general definition of finitarity: The preservation of directed colimits is equivalent to the preservation of filtered colimits (which is used in the definition in the nlab article). For the proof see the chapter "Directed and Filtered Colimits" in the book by Adámek and Rosický.
If you additionally require the $i\colon Y\to X$ to be a subset inclusion, you have the criterion of "finite boundedness" of a functor. In contrast to the definition in terms of filtered/directed colimits, you need a notion of finiteness in order to define finite boundedness of course. You have it in Set of course, but in a general category one usually talks about finitely presentable objects. However, the definition of finitely presentable is in terms of finitary functors. More precisely, an object $P$ is finitely presentable if the functor $\text{hom}(P,-)\colon $ is finitary. So we would run into a cyclic definition here.
In the article on finitary functors, the equivalence between finitarity and finite boundedness of functors between locally finitely presentable categories is proven generally (under some assumptions, and the article of course also provides both definitions you are asking for).
For the proof, there are two key-insights:
Even in Sets, one needs that $F$ preserves injective maps (but this is true up to re-definition of $F\emptyset$, keeping the rest of $F$ in tact).
For the direction finitary $\to$ boundedness, take the directed colimit $X = \textsf{colim}\{Y\subseteq X\mid Y\text{ finite}\}$, which is preserved by $F$. Hence $FX$ is a directed colimit, so $x \in FX$ must be an element $x\in FY$ for some finite subset $Y\subseteq X$ (Having understood colimits in set).
For the direction boundedness $\to$ finitary, take a directed colimit $D: (P,\le)\to \textsf{Set}$ with the colimit cocone $c_p\colon D_p\to C$, for $p\in P$. Since functors preserve composition, $Fc_p\colon FD_p\to FC$ is a cocone for the diagram $FD\colon (P,\le)\to \textsf{Set}$. We have to prove that this cocone is a colimit cocone. Since we know how colimits are done, one can prove that $FC$ does not identify too much and does not introduce new elements.
For the detailed/general proof see the linked arxiv article.
(Adámek, Jiří; Rosický, Jiří, Locally presentable and accessible categories, London Mathematical Society Lecture Note Series. 189. Cambridge: Cambridge University Press. xiv, 316 p. (1994). ZBL0795.18007. If you additionally require $i\colon Y\to X$ to be a subset inclusion, then the property you are giving is usually called finite boundedness.)