The set of points should be given by the second choice, i.e., the set of $\operatorname{Spec} \mathbb{C}$-points, over $\operatorname{Spec} \mathbb{C}$. However, there is an additional step you need to do before defining $\pi_1$ (besides choosing a basepoint), which is applying an analytification functor to endow the set with a suitable topology. This functor takes locally finite type schemes over $\operatorname{Spec} \mathbb{C}$ to complex analytic spaces. There is a brief exposition of analytification in SGA 1, Exp. 12, and in Serre's GAGA.
Regarding your comment about making a topological space from a variety defined over $\mathbb{Z}$, a $\mathbb{C}$-point of the base change to $\mathbb{C}$ over $\mathbb{C}$ is the same as a $\mathbb{C}$-point of the original scheme. This is the universal property of fiber product.
