What is the correct definition of an ind-scheme?
I ask this because there are (at least) two definitions in the literature, and they really differ.
Definition 1. An ind-scheme is a directed colimit of schemes inside the category of presheaves on the Zariski site of affines schemes, where the transition maps are closed immersions.
Definition 2. An ind-scheme is a directed colimit of schemes inside the category of sheaves on the Zariski site of affines schemes, where the transition maps are closed immersions.
This makes a difference, since the forgetful functor from sheaves to presheaves does not preserve colimits. For an ind-scheme $\varinjlim_n X_n$ according to Definition 1, we have $$\hom(Y,\varinjlim_n X_n) = \varinjlim_n \hom(Y,X_n)$$ for affine schemes $Y$. Does this also hold for non-affine schemes? Probably not. But one certainly wants the category of schemes to be a full subcategory of the category of ind-schemes. For this, Definition 2 seems to be more adequate.
Any detailed references are appreciated.