There are a few equivalent definitions of Stein manifolds. As far as I know they were initially defined as holomorphically convex complex manifolds, and then the other definitions (e.g. complex manifolds which embed holomorphically in a complex Affine space) were proved to be equivalent to the first afterwards. What I am missing is the motivation behind the first definition I mentioned. I know that for a domain in $C^n$, holomorphic convexity means it is a domain of holomorphy. But why was this definition generalized to a complex manifold, in other words, does holomorphic convexity perhaps imply anything about the complex functions on the manifold? or is a Stein manifold a domain of holomorphy in an ambient complex manifold?

There are a few equivalent definitions of Stein manifolds. As far as I know they were initially defined as holomorphically convex complex manifolds, and then the other definitions (e.g. complex manifolds which embed holomorphically in a complex Affine space) were proved to be equivalent to the first afterwards. What I am missing is the motivation behind the first definition I mentioned. I know that for a domain in $\mathbb{C}^n$, holomorphic convexity means it is a domain of holomorphy. But why was this definition generalized to a complex manifold, in other words, does holomorphic convexity perhaps imply anything about the complex functions on the manifold? or is a Stein manifold a domain of holomorphy in an ambient complex manifold?