History of terminology: sheaves, presheaves, etc.
- First let me congratulate you on your dictionary: it is absolutely correct and will be useful to other users as well.
- Yes, the terminology is essentially standardized and it is nice to keep the concepts of étalé space and sheaf distinct.
- Stalk and fiber are different concepts for locally ringed spaces, the arena for most of sheaf theory.
The most important examples of such locally ringed spaces $(X,\mathcal O_X)$ are differential manifolds, schemes and analytic spaces.
Given a sheaf $\mathcal F$ of $\mathcal O_X$-modules on $X$, we have at each $x\in X$ its stalk $\mathcal F_{x} $, an $\mathcal O_{X,x} $- module described by the inductive limit process you alluded to.
But we also have its fiber $\mathcal F(x)=\mathcal F_{x}/\mathfrak m_x\mathcal F_{x}$, a $\kappa (x)=\mathcal O_{X,x}/\mathfrak m_x$-vector space.
For example if $p:E\to X$ is a vector bundle on the real differential manifold $(X,\mathcal C^\infty _X)$, it has an associated sheaf of sections $\mathcal E$.
The stalk $\mathcal E_x$ is an infinite dimensional real vector space but the fiber $\mathcal E(x)$ is the finite-dimensional real vector space $\mathcal E(x)=p^{-1}(x)\subset E$ .
- Sheaf theory has an exciting, sometimes poignant, history starting with its invention in captivity by Jean Leray, a French officer made prisoner by the German military in WWII.
The changes in terminology result from the development of sheaf theory in various branches of mathematics in the hands of luminaries like Henri Cartan, Koszul, Serre, Godement, Grothendieck in the fifteen years following the end of the war.
Here and here are somewhat related posts on the history of the subject.