I was reading this page: https://www.encyclopediaofmath.org/index.php/Pseudo-group
The definition of a pseudo-group is on that page. I'll repeat it here.
Let $X$ be a topological space. Let $Open(X)$ denote the groupoid of open subsets of $X$ whose morphisms are homeomorphisms. A pseudo-group $G$ on $X$ is a subgroupoid of $Open(X)$, with two properties:
1) Pre-sheaf: If $W \subset U$ are opens, and $g : U \to V$ is in $G$, then $g|W$ is in $G$.
2) Gluing: If $W = \bigcup W_{\alpha}$ is a covering of an open set by open subspaces, and if $f : W \to W'$ is in $Open(X)$, then if $f|_{W_{\alpha}} \in G$ for all $\alpha$, then $f \in G$.
A pseudo-group is used to describe a manifold $M$ with extra structure, by requiring the transition maps between charts $\phi : U \to X$ are in $G$. In that context, one says that $M$ is a $G$-manifold. (Modulo details about maximal atlases, etc.)
For example, it is clear that if $X = \mathbb{C}$, then the (bi)holomorphic homeomorphisms between open subsets of $X$ defines pseudo-group $G$, and $G$-manifolds are Riemann surfaces.
The authors on that page (and a few other places) indicate that from a group $H$ acting on a space $X$, $\rho : H \to Aut(X)$, one can get a pseudo group $\Gamma(H)$ whose morphisms are the restriction of $\rho(g)$ to opens of $X$. It is clear that this satisfies all properties, except the gluing property.
However, it is not clear (and seems false), that this satisfies the gluing property. For example, let me take $G = GL_2(\mathbb{R})$, and $X = \mathbb{R}^2$, with the natural action. Let $U \subset X$ be the union of $B_{\epsilon}(1) =: B(1)$ with $B_{\epsilon}(-1) = B(-1)$, $\epsilon < 1/2$. Then I let $g_1$ be the action that rotates $B(1)$ a little bit counter-clockwise, and $g_2$ be the action that rotates $B(-1)$ a little bit clockwise. There is a homeomorphism $g : U \to g(U)$ that does $g_1$ and $g_2$ on each component, but this is clearly not a linear map. What am I missing? (Is there a secret sheafification ... though this would ruin the point of this concept...)
I'd like to understand this concept, which seems fundamental, so help would be appreciated. I'm sure I'm misunderstanding something...
For some additional context -- if we take $H= SO(n)$ with natural action on $S^n$, then manifolds modeled on $\Gamma(S^n)$ are supposed to be "spherical manifolds." $H = Isom(\mathbb{R}^n)$ produces "Euclidean manifolds. Note that groups have the property that if a homeomorphism $f : U \to V$ is locally in $H$, and $U$ is connected, then $f$ is in $G$. So in these cases the only sheafification that is necessary is across disconnected components of an open. This doesn't seem to be a major break with the philosophy of this definition.
(More generally, we only need to sheafify across disconnected components if $H$ has this property: $h_1, h_2 \in H$, and $U \subset W$ connected opens, then $h_1 | U = h_2 | U$ then $h_1 | W = h_2 | W$.)
Perhaps this is what is meant? Or am I missing something? Perhaps all $H$ that one encounters in practice have that property? (Any linear representation will.)
I think the example with $G=GL_2(\mathbb R)$ and $X=\mathbb R^2$ you are trying to make is not correct. If you want to use two balls with different centers, then there is no transformation in $G$ which rotates one of the balls but not the other. You can only consider global linear trsnaformations of $\mathbb R^2$ which map $U$ to itself and there are very few of those.
In general, I would say that the concept of a $G$-manifold as you describe is usually only used in the case of transitive pseudo-groups. Otherwise, you would have different classes of points in a $G$-manifold $M$ (corresponding to different $G$-orbits in $X$).
Edit (in view of your comment): I think you are mixing up two different concepts here. On the one hand, there is the concept of the pseudo-group defined by a group action. On the other hand, there is the concept of a $G$-manifold which makes sense once you have given a (transitive) pseudo-group. Let me explain this by an example in which both concepts are relevant (for a finite diemensional group, while most interesting examples of pseudo-groups are infinite dimensional). Let $G$ be the Euclidean group in dimension $n$ generated by translations and orthgonal maps and consider the natural action of $G$ on Euclidean space $X=\mathbb R^n$ (which is transitive). The pseudo-group generated by this is the subgroupoid of $Open(X)$ for which the morphisms from $U$ to $V$ are the restrictions of rigid motions which map $U$ to $V$. (So generically, there will be no morphisms from $U$ to $V$ at all.)
Now suppose that on a manifold $M$, you have charts with values in $X$ such that the transition functions are in this pseudo-group. Then you can pull bakc the flat Riemannian metric on each chart and this is preserved by chart changes by assumption. Hence you obtain a flat Riemannian metric on $M$. Converserly, given a flat Riemannian metric you get a $G$-structure.
In this example, the elements of the pseudo-group on $X$ are exactly the local isometries of the flat metric on $X$, so this coincides with the local automorphisms of the natural structure of $X$ as a $G$-manifold. However, this is only true by transitivity.
If you go to a non-transitive examples like the one in your question, these things fall apart (and I doubt that the concept of a $G$-manifold makes sense). The pseudo-group here is just given by restrictions of the actions of elements of $G$, and these are easily seen to satsify the gluing property. But such a restriction can never rotate a ball with center different from zero. Likewise, any transformation in $GL(2)$ preserves $0$, so any chart change has to map $0$ to $0$ for $G=GL(2)$. Thus in a $G$-manifold $M$, there would be two classes of points (those which are mapped to zero in one and thus in any chart and those which aren't) which leads to strange behavior.