sheaf theory - Definition of sheaves in wikipedia

I think you're quite right; Wikipedia's "concrete definition" is only correct for concrete categories whose underlying-set functor is (not just faithful but) conservative, i.e. such that any morphism which is a bijection on underlying sets is an isomorphism in the category. The page does say that the concrete definition "applies to the most common examples such as sheaves of sets, abelian groups and rings," all of which have this property, but it ought to be fixed to make clear in exactly what situations this definition applies.

Secondly, I observe that the "normalisation" condition in the Wikipedia concrete definition is also odd. Since the empty set is covered by the empty family, the "local identity" and "gluing" conditions already imply that the underlying set of $F(emptyset)$ is terminal. Saying that in addition, $F(emptyset)$ itself is terminal is an additional condition, which is in fact a special case of the second, more generally applicable, definition.

Thirdly, I think you're also right that for the correct general definition, the category doesn't need to have any limits a priori; you can just assert that $F(U)$ is the limit of the appropriate diagram of the $F(U_i)$ and $F(U_icap U_j)$.

Finally, let me go out on a limb and say that it seems to me that defining "sheaves with values in an arbitrary category" is often a misguided thing to do. More often, it seems like rather than "a sheaf with values in the category of X," the important notion is "an internal X in the category of sheaves of sets." For familiar cases such as groups, abelian groups, rings, small categories—in fact, for any finite limit theory—the two are the same, which may be what leads to the confusion. But the good notion of "sheaf of local rings," for instance, is not a sheaf with values in the category of local rings, but rather a sheaf of rings whose stalks are local (at least, when there are enough points), and that's the same as an internal local ring in the category of sheaves of sets. The situation is similar, I think, for "sheaves of topological spaces" (or locales). I'd be happy for people to point out where I'm wrong about this, though.