The eventual goal of this question is to better under the action of a Galois group on a scheme defined over a field $K$. Right now I just want to understand how to think about $K$-points and their Galois conjugates.
The $K$ points, or generally $R$-points ($R$ a ring) of a scheme $X$ are morphisms $\text{spec}\ R \to X$, or equivalently ring morphisms $K[X] \to R$, ($K[X]=\mathcal{O}_X(X))$. Suppose moroever $X$ is finite type and affine, for simplicity. Then the usual story explaining $X(R)$ is that they are the intuitive "$R$-valued points" that satisfy the solutions to equations defining $X$.
Recently when trying to define/understand the Galois action of schemes I realized for $K$-points of a Scheme over $K$, there are more $K$-points than I previously thought: for the simplest example, take $\mathbb{A}^1_K$: writing points as morphisms $K[x] \to K$:
$$c_0+c_1x+\cdots c_mx^m \mapsto c_0+c_1 \alpha + \cdots + c_m \alpha ^m $$ $$c_0+c_1x+\cdots c_mx^m \mapsto c_0+c_1 (\sigma\alpha) + \cdots + c_m(\sigma \alpha) ^m$$
This is what I would think of is the natural meaning of a Galois action on $K$-points. However I also can think of two other maps:
$$c_0+c_1x+\cdots c_mx^m \mapsto \sigma c_0+\sigma c_1 \alpha + \cdots + \sigma c_m \alpha ^m $$
$$c_0+c_1x+\cdots c_mx^m \mapsto \sigma c_0+\sigma c_1 (\sigma \alpha) + \cdots + \sigma c_m (\sigma \alpha ^m) $$
The first pair of maps is what I would naively call the $\sigma$ action on $K$-points. However, I am not sure it is the correct notion. My basic question is, which (pair) of theses maps deserves to be identified with the pair of points $(\alpha, \sigma \alpha)$. The only straightforward relation I see among these maps is that map 4 can be obtained from map $1$ by post-composing with $\sigma$. But they have the same kernels, which I don't know is relevant of not. What is the proper way to think about the relationship between these different maps?
(I am aware that a bunch of these maps can be cut out by working in $\text{Sch}_{/K}$ and requiring the maps to commute with the respective structure morphisms. I hesitate to impose this because I remember reading that the Galois action on schemes over fields does not respect the structure morphism. If I am mistaken, comments correcting my thoughts in this direction would be helpful as well).