I encountered phrases such as "sheaf of germs of continuous functions" (e.g., page 164 of the book "Foundations of Differentiable Manifolds and Lie Groups" by Warner 1983) and "sheaf of continuous functions" (e.g., this wiki page). Are they the same concept?
Specifically, we may define the "sheaf of continuous functions" on a manifold $M$ as the contravariant functor that (a) associates any open subset $U\subseteq M$ with a set/vector-space/algebra $\mathcal C^0(U)$ of continuous functions on $U$, and (b) associates each inclusion of open subsets $U\subseteq V$ with the restriction map $\mathcal C^0(V)\to\mathcal C^0(U)$. On the other hand, we may define the set $\mathscr C^0(M)$ of germs (in other words, the union of all stalks), made into a topological space, which is called the "sheaf of germs of continuous functions", although I don't see how the topological space is a contravariant functor.
My questions: are these the same concept, or are they related but not conceptually the same, or are they completely un-related? Also, how are these two concepts related to the notion of Etale spaces? Is the "sheaf of germs of continuous functions" synonymous to the Etale space? Also, I read about (but haven't quite understood) an equivalence between "categories of Etale spaces" and "categories of sheaves". Does that play a role in this story?
I'm a theoretical physics student and I'm very new to this level of abstraction. The above is what I'm able to gather so far, and I have managed to understand most of these definitions, by taking a "theoretical minimum" approach. However, I know very little beyond these definitions, so I would certainly appreciate spelling out properties/theorems explicitly.
Many thanks in advance!