A very elementary question on the definition of sheaf on a site

That exactness conditions can be rephrased more explicitely as:

$$ Hom(V,Z) = \left\lbrace (v_i) \in \prod_i Hom(U_i,Z) \ \middle| \ \forall i,j,v_i \circ \pi_1 = v_j \circ \pi_2 \right\rbrace $$

where $\pi_1,\pi_2$ denotes the two projections $U_i \times_V U_j \rightrightarrows U_i,U_j$.

When you write it like this, the condition in the case $i=j$ is clearly vacuous when all the map $U_i \to V$ are monomorphisms. Indeed in this case $U_i \times_V U_j$ is justs the intersection of $U_i$ and $U_j$, so that $\pi_1=\pi_2$ when $i = j$. This case is very frequent, and you can very often restrict to it by considering the "image" of the $U_i$ in $V$.

But in some situation (for e.g. if you want to keep your objects $U_i$ be to in some specified site that do not admit image factorization like the étale site) it might not be the case. and in general you need the case $i=j$. Consider the case where the you only have a single map $U \to V$. Then the condition becomes

$$ Hom(V,Z) = \left\lbrace f \in Hom(U,Z) \ \middle| \ f\circ \pi_1 = f \circ \pi_2 \right\rbrace $$

where $\pi_1$ and $\pi_2$ are the two projections $U \times_V U \rightrightarrows U$.

You can think of $U \times_V U \rightrightarrows U$ as a map $U \times_V U \to U \times U$ which is a monomorphisms and corresponds to the "equivalence relation such that $V$ should be the quotient of $U$ by this relations". Or rephrased this as $V$ being the coequalizer ('in the category of sheaves') $U \times_V U \rightrightarrows U \to V$, i.e. $V = U /R$ where $R$ is the equivalence relation $U \times_V U$.

And a function from $V \to Z$ can be described as a function $U \to Z$ which is compatible to the equivalence relation $R$ such that $U/R \simeq V$.

Also note that in the general case (with several map) you can think of the general condition as being in two part: you have the condition for $i=j$ that assert that each maps $U_i \to Z$ factors through "the image $V_i$ of $U_i$ in $V$" (if this make sense) , and the condition for $i \neq j$ that implement the usual compatibility condition.

Yes, you have to include the case $i=j$. Just look at what happens in the case of a single $U_1$, in order for this to boil down to the concept of a single effective epimorphism (introduced in the preceding paragraph in the book) you need to include $i=j=1$.