The following discussion is summarized from the paper "Finite flat group schemes" by Tate.
Let $S$ be a non-empty connected scheme and $\alpha:\operatorname{Spec}(\Omega)\rightarrow S$ a geometric point centered at $s\in S$, that is a field embedding $\tilde{\alpha}:\kappa(s)\rightarrow \Omega$ of the residue field of $s$ into an algebraically closed field $\Omega$.
There is a natural functor from the category of finite étale schemes $Y$ over $S$ to the category of sets, sending $Y$ to the set $Y(\alpha)$ of geometric points of $Y$ mapping to $\alpha$. An element of $Y(\alpha)$ is an embedding $\tilde \beta:\kappa(y)\rightarrow \Omega$ where $y\in Y_s = Y\times_S\{s\}$ is a point of $Y$ over $s$ and such that the restriction of $\tilde \beta$ to $\kappa(s)$ is precisely $\tilde \alpha$.
The fundamental group $\pi=\pi_1(S,\alpha)$ of $S$ over the geometric point $\alpha$ is defined as the automorphism group of this functor. More concretely, an element $\sigma\in \pi$ is a family $\sigma_Y$ of permutations of the sets $Y(\alpha)$ for each finite étale scheme $Y$ over $S$, which behave well with morphisms $Y\rightarrow Y'$.
In the paper, it is claimed that « if $\alpha'$ is another geometric point of $S$, the functors $Y\mapsto Y(\alpha)$ and $Y\mapsto Y(\alpha')$ are isomorphic [...] it induces an isomorphism $\pi_1(S,\alpha)\cong\pi_1(S,\alpha')$».
In this last claim, is it implied that $\alpha$ and $\alpha'$ are still centered at the same point $s\in S$ or not ?
If we could consider geometric points with different centers, their respective residue field may be very different. For instance, if $S=\operatorname{Spec}(\mathbb Z_p)$ is the spectrum of the ring of $p$-adic integers, I can consider a geometric point $\operatorname{Spec}(\overline{\mathbb Q_p})\rightarrow S$ over the zero ideal of $\mathbb Z_p$, and the geometric point $\operatorname{Spec}(\overline{\mathbb F_p})\rightarrow S$ over the maximal ideal $(p)$ of $\mathbb Z_p$. Both fields having different characteristics, I would expect the associated functors $Y\mapsto Y(\alpha)$ to be very different... Is it true ?