The definition of a "connected Shimura datum" (as in Milne's notes) is a pair $(G, X)$, where $G$ is a reductive algebraic group and $X$ is a $G(\mathbb{R})$-conjugacy class of morphisms $$ x: \mathbb{S}^1 \to G_\mathbb{R}, $$ where $\mathbb{S}^1$ is the norm one subtorus of $\text{Res}_{\mathbb{C}/\mathbb{R}}$ satisfying a short list of axioms. Given such a morphism $x$, one gets a family of morphisms $$ x_n: \text{Res}_{\mathbb{C}/\mathbb{R}} \mu_n \to G_\mathbb{R} $$ compatible with the natural inclusions $\mu_n \hookrightarrow \mu_{mn}$, and if I'm not mistaken, by Zariski density, $x$ is determined uniquely by the $x_n$, and moreover, two maps $x$, $x'$ are $G(\mathbb{R})$ conjugate if and only if their associated families $x_n$, $x_n'$ are. It also makes sense to demand that Deligne's axioms hold for the $x_n$, and one sees that if they hold for $x$, they hold for $x_n$.

From the perspective of special points and canonical models, it is not clear to me where one uses the morphism $x$; on the level of real points, it seems that only the action of roots of unity are used, and so much of the theory should be recoverable from only the $x_n$; however, there is a lot of it that I haven't understood yet.

My questions are:

if we define a generalized Shimura datum to be a family of $x_n$ compatible with the natural inclusions (equivalently, a conjugacy class of maps from the direct limit of the $\mu_n$), do we get generalized Shimura varieties?

If so, are there generalized Shimura varieties which do not come from Shimura varieties?