I admit I am a bit confused by your terminology, but here is how I learned it: Let $P$ be a $G$-principal bundle and $\Sigma$ a spacetime.
- gauge group: The fibers of the $G$-principal bundle over the spacetime, i.e. the group $G$.
- (Local) group of gauge transformations: The group of diffeomorphisms $t : P \rightarrow P$, which are fiber-preserving and $G$-equivariant, i.e. if $\pi : P \rightarrow \Sigma$ is the projection then $\pi \circ t = \pi$, and $t$ commute with the group action on $P$.
One can now, by transitivity of the group action on the fibers, define a function $g_t: P \rightarrow G$ by $t(p) = pg_t(p) \forall p \in P$, and such functions $g : P \rightarrow G$ conversely define a gauge transformation by $t_g(p) = pg(p)$ as long as they fulfill $g_t(ph) = h^{-1}g_t(p)h \forall h \in G$, so we have two alternative characterizations of local gauge transformations:
$\mathcal{G} = \{t |t \in \mathrm{Diff}(P) \wedge \pi \circ t = t \wedge t(ph) = t(p)h \forall h \in G\} = \{ g| g \in \mathrm{Maps}(P,G) \wedge g(ph) = h^{-1}g(p)h \forall h \in G \}$
The equivariant diffeomorphisms of $P$ are called local, since they apply a different group element to every spacetime point.
Now, the associated bundles are affected as follows: Let $\phi : \Sigma \rightarrow P \times_G V$ be a section of the associated bundle, i.e. a field. By a similar argument to the above, these are in bijection to $G$-equviariant functions $f_\phi : P \rightarrow V$ satisfying $f_\phi(pg) = \rho(g^{-1})f_\phi(p)$. This is esentially the reason why, in $\mathrm{U}(1)$ symmetry, a gauge transformation $\mathrm{e}^{\mathrm{i}\alpha(x)}$ acts on fields as $\phi(x) \mapsto \mathrm{e}^{-\mathrm{i}\alpha(x)} \phi(x)$.
So, you see, the local group of gauge transformations is much bigger that the global gauge group since it allows far more functions than just the constant ones. You can always clearly write down the global gauge group (it defines your theory!), but writing down the local one more explicit than I did above is hard. For $\mathrm{U}(1)$, however, it is just $\{x \mapsto \mathrm{e}^{\mathrm{i}\alpha(x)} x | \alpha : P \rightarrow \mathrm{U}(1) \text{is smooth (enough)}\}$, I think. Cases where the two groups coincide demand a spacetime that is a point, I would guess, but I am not wholly confident in that.
Also, all of this can be done classically, nothing about gauge theories is inherently quantum.
EDIT:
Alright, your edit was very helpful in discerning what is actually going on here.
Your global gauge group is what physicists call the group of gauge transformations. The gauge group of a a gauge theory is what you call a local gauge group (and what the nLab also calls the local gauge group). When physicists say the gauge group $\mathrm{SU}(N)$, they mean it is what you call the local gauge group.
The global gauge group of the nLab is just the group of transformations (not necessarily gauge transformations, terminology is terrible here, I know) that leaves all observables invariant, i.e. it is the group of symmetries of the theory (not the group of symmetries of the Lagrangian), the group of gauge transformations is naturally a subgroup of this. The difference is that this global gauge group can contain transformations that have not really something to do with the structure of the local gauge group, and can contain things which are not gauge transformations. This global gauge group can even exist if you have no explicit gauge theory, and is inherently a QFT concept.
In other news, you are right, your connection form $\omega$ is the gauge field $A$ of a physical gauge theory, and it transforms exactly like you wrote. Now, the problem with the gauge field is exactly that ugly transformation, so we construct the curvature transforming in the adjoint rep and call it the field strength $F$. The action of a pure (Yang-Mills) gauge theory is then (up to prefactors) given by
$$ \int_\Sigma \mathrm{Tr}_{ad}(F \wedge \star F)$$
since the action must be invariant under gauge transformations and the $\mathrm{Tr}_{ad}(F \wedge \star F)$ is pretty much the only object we can construct out of the gauge fields that is invariant and can be integrated over the spacetime.
After a bit of thought, I have to come to some sort of understanding that I hope will be useful in making the ideas a bit more precise. In gauge theories physicists begin with a Lagrangian $ \mathcal{L}[\phi,\dot{\phi}] $. The claim is that this $ \mathcal{L} $ is invariant under the action of some group. What is a bit tricky is that to make this statement a bit more precise requires that we have two fiber bundles at once namely the principal-$G$ bundle and its associated vector bundle. I will try to describe things in such a way that the mathematical structure emerges rather than merely providing a dictionary.
1.For each patch of spacetime,$\mathcal{U}_i$ where $\mathcal{M}$ is base manifold we pick a map $S: \mathcal{U}_i \rightarrow G $ $(\textit{this will later be the gauge group})$. Then we pick a certain representation of the group i.e $\rho: G \rightarrow V $ where $V$ is a vector space. We now define what we later be a section $ \psi: \mathcal{U}_i \rightarrow [x, \phi] $ where $x$ is a point on the manifold. Gauge invariance means that $[x,\phi] \sim [x,\rho(g^{-1})\phi] $. We will come back to this construction but I think it is better at this point to concentrate on the map $S$ at this point
2.So rememeber we are at the spacetime patch $\mathcal{U}_i$ with the map $S$ in our hand. With this we construct the cartesian product $ \mathcal{U}_i \times G $. If we happen to find two overlapping open sets $ \mathcal{U}_i \text{ and } \mathcal{U}_j $ then for the sets of points in the intersection we have to make sure things are consistent and so we define functions $ t_{ij}: \mathcal{U}_i \cap \mathcal{U}_j \rightarrow G $ that will act on G i.e $(x, G) \rightarrow (x,t_{ij}(x)G)$. Doing this for the whole manifold $\mathcal{M}$ gives us another manifold $\mathcal{P}$ that is locally $\mathcal{U}_i \times G $. This the principal-$G$ bundle.
3.As we all know, requiring local gauge invariance in step one meets a glitch. The problem is the map $S$ because as we go around on the manifold $\mathcal{M}$ we need a method to go from one fiber to another fiber on the principal-$G$ bundle. To do this requires we introduce a connection $\Omega$ on the principal-$G$ bundle. But physicists always work on the base manifold so we need to pull back $\Omega$ to the base manifold by some section $\sigma $ i.e calculate $\sigma^*\Omega \equiv A $. Now of course these are locally defined sections and so when we are in intersection of two spacetime patches $\mathcal{U}_i\cap \mathcal{U}_j $ we will have two sections $\sigma, \sigma'$. This means we will get $ \sigma^*\Omega = A $ and $\sigma'^*\Omega=A' $. The two sections are related by the map $S$ i.e $\sigma' = R_{S(x)}\sigma = \sigma S(x) = \sigma g $ i.e the group acts by a right action.
4.To calculate $\sigma^*\Omega$ we note that $<\sigma^*\Omega, v>= <\Omega, \sigma_* v >$ where $v$ is a vector on the principal bundle.
A tricky calculation shows $\sigma'_*v = R_{g*}(\sigma_{*}v)+\eta_x(p) $ where $\eta_X$ is a fundamental vector field, $X = <S^*\theta,v>$ and $\theta $
is the Maurer-Cartan form and $g$ is the image of $S$. To show this works we do the calculation
$(\sigma'^* \omega)(v)= <\Omega,\sigma'v> = <\Omega,R_{g*}(\sigma_*v)>+<\Omega,\eta_X(pg)= <R^*_g\Omega,\sigma_*v> +X = <Ad_{g^-1}\Omega,\sigma_*v>+X=<Ad_{g^{-1}}A + S^*\theta,v> $ This of course is the usual transformation rule for the gauge field on the base manifold
5.We can now state the fact that we have an associated bundle $\mathcal{A}$ which is $\mathcal{P}\times_G V = (\mathcal{P}\times V)/G $ and is locally $\mathcal{U}_i \times V $ with sections as defined in step 1. The sections on this bundle are what physicists call the fields.
Best Answer
Let's consider the torus glued from two cylinders $C_1$ and $C_2$ with trivial spin bundles over them. We know there are four different spin bundles over the torus, how do we get them from choosing a transition function $t_{12} : U_1\cap U_2 \to \mathrm{Spin}$ on the two (thickened) circles where the cylinders overlap? Let's split this into two "partial" transition functions $t_1,t_2 : S^1 \to \mathrm{Spin}$. The choice for the trivial bundle on the torus is clearly $t_1 = t_2 = 1$. A second bundle arises from choosing $t_1 = 1; t_2 = -1$:
Bundles constructed from patches via gluing are isomorphic if there are functions $f_i : C_i \to G$ on the glued parts such that $t_{12} f_2 = f_1 t_{12}$ on the overlap. Since the $C_i$ are connected but 1 and -1 lie in different connected parts of $\mathrm{Spin}(1,1)$, there are no smooth functions such that $f_1 = f_2$ on one circle and $f_1 = -f_2$ on the other, therefore the bundles for $t_1 = t_2 = 1$ and $t_1 = 1; t_2 = -1$ are non-isomorphic.
Now if we look at the spinor field $\psi$ (really a section of some associated bundle to the spin bundle) in the patches $C_i$, then we have two fields $\psi_i$ with $\psi_1 = t_{12} \psi_2$. Clearly, in the trivial case, this is just $\psi_1 = \psi_2$ and so we can pretend we have a spinor field with periodic conditions. In the case where the $t_i$ have different signs, we get that our $\psi_i$ agree on one of the overlap circles but have $\psi_1 = -\psi_2$ on the other - so we have an "antiperiodic boundary condition" going around the fundamental loop orthogonal to the overlapping circles.
If we choose the two $C_i$ in "the other direction", i.e. if we glue the torus together longitudinally instead of latitudinally, we get a another non-trivial bundle. Combining both of these gluing methods (yes, this is handwavy but you should be able to make it precise considering the actual minimal four charts on the torus) gives us four different bundles: Trivially glued along both fundamental loops, non-trivially glued along either one of them and non-trivially glued along both of them, and they corresponding to the "boundary conditions" for spinors as described above.
(That this exhausts all possible bundles, i.e. that there are $2^{2g}$ possible bundles is proven by connecting spin bundles to the algebraic geometry of Riemann surfaces, see Atiyah's "Riemann Surfaces and Spin Structures".)
This reasoning extends to all Riemann surfaces by viewing them as the $g$-fold connected sum of the torus with itself.