I'm going to write down the definitions as from The Rising Sea: Foundations of AG – Nov 18 2017 draft, starting from 2.2.3. I feel that the definition of a germ is circular.
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The setting: We have a topological space $(X, \tau)$ and a function $F: (U: \tau) \rightarrow \operatorname{Diff}(U)$, which assigns to each open set $U \in \tau$, the set of differentiable functions over $U$.
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Sections of a presheaf $F$ over an open set $U$:
For each open set $U \in \tau$, we have a set $F(U)$. The elements of $F(U)$
are called as the sections of $F$ over $U$. -
Restriction Map: For each inclusion $U \hookrightarrow V$ ($U \subseteq V$),
we have a restriction map $Res(V, U): F(V) \rightarrow F(U)$. -
Identity Restriction: The map $Res(U, U)$ is the identity map.
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Restrictions Compose: If we have $U \subseteq V \subseteq W$, we must have
$Res(W, U) = Res(W, V) \circ Res(V, U)$. -
Germ at a point $p$ (1): A germ of a point $p$ is any section over any open set $U$ containing $p$.
That is, the set of all germs of $p$ is formally $\operatorname{Germs}(p) \equiv \{ F(U_p) : p \in U_p \in \tau \}$.
We sometimes write the above set as $\operatorname{Germs}(p) \equiv \{ (f, U_p) : f \in F(U_p), p \in U \in \tau \}$.
This way, we know both the function $f$ and the open set $U_p$ over which it is defined. -
Stalk at a point $p$: A stalk at a point $p$, denoted as $F_p$,
consists of equivalence classes of all germs at a point, where two germs are
equivalent if the germs become equal over a small enough set.
We state that $(f, U) \sim (g, V)$ iff there exists a $W \subseteq U \cap V$ such that
the functions $f$ and $g$ agree on $W$: $Res(U, W)(f) = Res(V, W)(g)$. -
Germ of $f$ at $p$ (2): If $p \in U$ and $f \in F(U)$, then the image of $f$ in $F_p$, as in, the value
that corresponds to $f$ in the stalk is called as the germ of $f$ at $p$.
This last definition doesn't make sense. We have already defined the germ at a point $p$ (1) before. Now we are re-defining the germ at point $p$ with definition (2). The definition (2) is an equivalence class of elements of definition (1). So when someone says "germ", which definition do they really mean? This feels quite circular.
Best Answer
First off, the germs in Vakil’s notes, as far as I can tell, are defined not just as tuples of open sets and functions on them, but as such tuples modulo a certain equivalence relation, that is: as equivalence classes. This also how it is used by everyone. Germs are the elements of stalks.
Anyway: It’s not strictly re-defining the notion of a germ. First, a germ at point $p$ is defined (and I think it’s better to use “at” rather than “of”). Next, the germ of function $f$ at point $p$ is defined – notice it says: the germ … of a function. As we expect, the germ of a function $f$ at a point $p$ is indeed a germ at $p$. This very much akin to the definition of a residue class (as an element of a factor structure) as opposed to the definition of the residue class of an element (as the image of an element under the residue class projection).
(Answer adapted from comments.)