TITLE: Bundle Automorphisms, Structure Groups and Gauge Groups
QUESTION [6 upvotes]: I am trying to get my head around the mathematical foundations of gauge theory and wanted to check that I am correct in thinking the following is true.

If $E$ is a $G$-principle bundle over $M$ then we call $G$ the structure group of $E$ and it is the group of transition functions at any point $x \in M$.
The group of bundle automorphisms is precisely $G$ at any point $x \in M$.
We call the group of bundle automorphisms the group of (global) gauge transformations. 
We call $G$ the gauge group.
We call the group of transition functions on some neighbourhood $U$, the group of (local) gauge transformations. It is the group of bundle automorphisms over $U$.

I feel that I may be mixing up global and local notions here a bit, particularly with regard to automorphisms and transition functions. Is there a better way to phrase the interaction between the group of transition functions and the automorphism group?
I'd appreciate mathematical precision, since the hand-wavy arguments in physics books are precisely what confuses me on these points! Many thanks in advance!

REPLY [5 votes]: This is not really an answer. It seems that I can't add a comment (low reputation). Just wanted to correct a small (but crucial) inaccuracy in Henry T. Horton's fine answer: in general $\mathscr{G}$ can be identified with $C^\infty(M,G)$ only when $G$ is Abelian or the principal bundle $E$ is trivializable. See e.g. Husemoller, Fibre bundles, Proposition 1.7 p. 81.