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.
Best Answer
That's just the active/passive distinction you have for all transformations: Is a "rotation" an actual, physical motion of an object or is it rotating your coordinate system? It depends on the context - and often it doesn't actually matter!
There's nothing special about gauge transformations in this context, sometimes people will mean the active version and sometimes they will mean the passive version and sometimes they won't have thought about which they mean at all.
However, in terms of "bundle theory" it is not always sufficient to think about gauge transformations as happening on a fixed principal bundle: There are "gauge transformations" in physics that change the bundle and that's one meaning of the phrase "large gauge transformation", see this question and this question.