The definitions are equivalent as stands; no extra conditions (eg majorants for infinite sets of factorisations) are needed. This is essentially because colimits in $\mathbf{Sets}$ are computed finitarily.

One way to present the colimit $\varinjlim_i Hom(X,F(i))$ (see eg Mac Lane CWM) is as

1. the coproduct of all the sets $Hom(X,F(i))$, quotiented by

2. the relation $\sim$ defined by: for $f \colon X \to F(i)$ and $g \colon X \to F(i')$, take $f \sim g$ if there is some $i''$ and a commutative square as in your original post.*

Looking at it this way, the two conditions in Makkai and Paré’s definition say that the canonical map $\varinjlim_i Hom(X,F(i)) \to Hom(X,\varinjlim_i F(i))$ is

1. surjective;

2. injective;

so together they say exactly that it’s an iso, which is what the usual definition says.

* For a general colimit, we’d need to use “the equivalence relation generated by $\sim$” (which is still someting finitary), but if the colimit is filtered, so a fortiori if it’s $\kappa$-filtered, then $\sim$ is already an equivalence relation.

The definitions are equivalent as stands; no extra conditions (eg majorants for infinite sets of factorisations) are needed. This is essentially because colimits in $\mathbf{Sets}$ are computed finitarily.

One way to present the colimit $\underset{i}{\varinjlim} Hom(X,F(i))$ (see eg Mac Lane CWM) is as

1. the coproduct of all the sets $Hom(X,F(i))$, quotiented by

2. the relation $\sim$ defined by: for $f \colon X \to F(i)$ and $g \colon X \to F(i')$, take $f \sim g$ if there is some $i''$ and a commutative square as in your original post.*

Looking at it this way, the two conditions in Makkai and Paré’s definition say that the canonical map $\underset{i}{\varinjlim} Hom(X,F(i)) \to Hom(X,\underset{i}{\varinjlim} F(i))$ is

1. surjective;

2. injective;

so together they say exactly that it’s an iso, which is what the usual definition says.

* For a general colimit, we’d need to use “the equivalence relation generated by $\sim$” (which is still something finitary), but if the colimit is filtered, so a fortiori if it’s $\kappa$-filtered, then $\sim$ is already an equivalence relation.