Mazur's theorem on the torsion points of an elliptic curve includes the fact that each possibility which can occur does in fact occur for infinitely many rational elliptic curves. I've read that this is proven by showing $X_1(n)$ is rational and has a rational point, thereby showing it has infinitely many rational points.
To be a rational point of $X_1(n)$ is the same as giving a morphism $\mathrm{Spec}(\mathbb{Q}) \rightarrow X_1(n)$. This in turn induces an elliptic curve $E \rightarrow \mathrm{Spec}(\mathbb{Q})$. But is $E$ necessarily defined over $\mathbb{Q}$? For example, $E : y^2 = x^3 + x + i$ has a morphism to $\mathrm{Spec}(\mathbb{Q})$, as does every other elliptic curve. Simply compose the morphism $E \rightarrow \mathrm{Spec}(K)$ with the morphism $\mathrm{Spec}(K) \rightarrow \mathrm{Spec}(\mathbb{Q})$ where $K$ is the field of definition.
I am clearly missing something here. How does giving $\mathrm{Spec}(\mathbb{Q}) \rightarrow X_1(n)$ guarantee that the resulting elliptic curve is defined over $\mathbb{Q}$? More generally, is a similar statement true for general moduli, or is there something special about elliptic curves?