In my specific case, $G=\mathrm{Spec}(k[M])$ is an algebraic torus acting on a toric variety $X_\Sigma$ corresponding to a fan $\Sigma$ when $k$ is not necessarily algebraically closed (or maybe even $k=\mathbb{Z}$). I see how one still can define the distinguished points $x_\sigma$, but how would the $G$-orbit of $x_\sigma$ be defined?
My problem is that my working definition of orbits uses points. Of course, I can consider $T$-valued points for any $T$ and get orbits of those, but how do they fit together?
If you have a group scheme $G\to S$, acting on an $S$-scheme $X$, then for any $T$-valued point $\rho : T\to X$ of $X$, the orbit of $\rho$ can be defined as the image of the natural map $$ G_T=G\times_S T \stackrel{1 \times \rho}{\to} G\times_S X \stackrel{\mu}{\to} X$$ where $\mu$ defines the action of $G$ on $X$. The orbit usualy is not closed (constructible under mild conditions), but one can consider the Zariski closure of the orbit, this is called orbit closure in Fulton's book on toric varieties.