In Vakil's book (Foundation of Algebraic Geometry), I learned the definition of a type of morphism of schemes being local on the target:
A class of morphisms of schemes is called local on the target if
(a) if $\pi:X\rightarrow Y$ is in the class then for any open set $V$ of $Y$, the restricted morphism $\pi ^{-1} (V)\rightarrow V$ is in the class.
(b) for a morphism $\pi:X\rightarrow Y$, if there is an open cover {$V_i$} of $Y$ for which each restricted morphism $\pi ^{-1}(V)\rightarrow V$ is in the class, then $\pi$ is in the class.
In the other book (Algebraic Geometry and Arithmetic curve Qing Liu), I learned a similar definition.
A property P of morphisms of schemes $\pi:X\rightarrow Y$ is said to be local on the base if the following assertions are equivalent:
(i) $\pi$ verifies P.
(ii) for any $y\in Y$, there exists an affine neighborhood $V$ of $y$ such that the restricted morphism $\pi ^{-1} (V)\rightarrow V$ verfies P.
It's obvious that "local on the target" implies "local on the base", but do we have another direction?
Local on the base implies condition (b), but not condition (a), of local on the target.
For (b): Suppose $(V_i)_{i\in I}$ is an open cover of $Y$ such that $\pi_i\colon \pi^{-1}(V_i)\to V_i$ satisfies property $P$. By (i) implies (ii), for all $i\in I$ and all $y\in V_i$, there is an affine neighborhood $U_{i,y}$ with $y\in U_{i,y}\subseteq V_i$ such that $\pi_{i,y}\colon \pi^{-1}(U_{i,y})\to U_{i, y}$ satisfies $P$. But then $(U_{i,y})_{i\in I, y\in V_i}$ is an affine open cover of $Y$ such that the restrictions $\pi_{i,y}$ satisfy $P$, So by (ii) implies (i), $\pi$ satisfies $P$.
For a counterexample to (a), you can take the property $P$ to be "$Y$ can be covered by affine opens, each isomorphic to $\mathbb{A}^1$." It's easy to see that this property is local on the base. But it does not satisfy condition (a) of local on the target. Indeed, the identity map $\mathbb{A}^1\to\mathbb{A}^1$ satisfies $P$, but letting $V$ be the open subset of $\mathbb{A}^1$ obtained by removing a point, the restriction $V\to V$ does not satisfy $P$, since $V$ doesn't have any open subsets isomorphic to $\mathbb{A}^1$.
This property $P$ seems pretty pathological to me (and not really "local"), which makes me feel like Vakil's definition is "better" for ruling it out. But I'm not an algebraic geometer, so I can't comment on whether the distinction between these definitions is relevant or just an oversight.