In Liu's Proposition 3.2.18, it is claimed that given an affine $k$-variety $X = \text{Spec}(k[X_1,\dots,X_n]/I)$ and a field extension $K/k$, we can identify the set $X(K)$ of $K$-rational points (or $K$-valued) and the solution space $\{(x_1,\dots,x_n)\in K^n\mid P(x_1,\dots,x_n)= 0 \text{ for all $P \in I$}\}$ over $K$.
This can be shown by noting that a $k$-morphism $$\text{Spec}K \to X$$ is equivalent to a $k$-algebra homomorphism $$k[X_1,\dots,X_n]/I \to K,$$ which is in bijection with a point in the solution space over $K$ via the images of $X_i$ in $K$.
Now, my question is how to describe the closed points in $X(K)$. Here is my attempt. We know that a closed point of $X$ corresponds to a surjection $$\pi: k[X_1,\dots,X_n]/I \twoheadrightarrow L$$ whose kernel is the corresponding maximal ideal, where $L/k$ needs to be a finite field extension by Hilbert's Nullstellensatz. In this case, to get a similar bijection we need to consider an equivalence relation on the set of such $k$-algebra surjections onto $L$ defined by $\pi\sim \pi'$ if and only if $\text{ker}(\pi) = \text{ker}(\pi')$. This equivalence relation is same as the equivalence given by the $\text{Aut}(L/k)$-action on the target. Therefore, I believe that for a finite field extension $K/k$, closed points in $X(K)$ is in bijection with the set $$\{(x_1,\dots,x_n)\in K^n\mid P(x_1,\dots,x_n)= 0 \text{ for all $P \in I$}\}/\text{Aut}(K/k)$$ where $\text{Aut}(K/k)$ acts diagonally on $K^n$, which I believe is then in bijection with $\text{Aut}(K/k)$-orbits of $X(K)$. Now, for example if $X= \text{Spec}\mathbb R[t]$, then $X(\mathbb C)$ is in bijection with $\mathbb C$ while closed points in $X(\mathbb C)$ is in bijection with $\mathbb C/ \text{Gal}(\mathbb C/\mathbb R)$, so this is not a canonical description of closed points in $X(\mathbb C)$ identified with $\mathbb C$. However, this cannot be avoided since $\mathbb C$-rational closed points of $X$ correspond to irreducible quadratics and hence there is no way to canonically identify it with a point in $\mathbb C$ (and the choice should be up to $\text{Gal}(\mathbb C/\mathbb R)$ as desired). This sheds light on the fact that a bijection $X(K)$ with the solution space over $K$ is not canonical and choices are up to the action of $\text{Aut}(K/k)$.
In summary, my question is whether my understanding/attempt above is correct. Also, as an aside question, the preceding argument, if correct, gives a bijection between closed points in $X$ with the disjoint union of $\text{Aut}(K/k)$-orbits of $X(K)$ for every isomorphism class of finite extensions $K/k$. Is there a description of closed points that you think is better than this? Thank you in advance.
I found the result I wanted to make sense of in Poonen's Rational points on varieties (Proposition 2.4.6). Let $X$ be a $k$-variety. Then, the map $$\{\text{$\mathfrak{G}_k$-orbits in $X(\overline{k})$}\} \to \{\text{closed points of $X$}\}$$ sending orbit of $(f:\text{Spec}\overline{k}\to X)$ to $f(\text{Spec}\overline{k})$ is a bijection. Here, $\mathfrak G_k$ denotes the profinite group $\text{Gal}(k_s/k)$ for the separable closure $k_s$ of $k$ (or equivalently $\text{Aut}_k(\overline k)$).