Let $A$ be a commutative ring. I thought that an exact functor (from the category of $A$-modules to itself) is defined to be a functor which sends every exact sequence to an exact sequence. But many books seem to define it to be a functor which sends every short exact sequence to a short exact sequence. But I don't think they are equivalent unless it sends a zero module to a zero module.
Is it true that every functor on the category of $A$-modules to itself sends a zero module to a zero module?
On
An exact functor is additive. It can be shown that the condition that a functor is additive is equivalent to the condition that it preserves the zero object and binary direct sums; see here. Hence, any exact functor preserves the zero object. It is not necessary that every such functor the zero object. If you consider the $\mathrm{Ab}$-enriched category $R\mathrm{Mod}$ with finite firect sums, then you get the fact that exact functors from $R\mathrm{Mod}$ to $R\mathrm{Mod}$ take the zero module to itself.
On
The "correct" definitions (in that they generalize the most smoothly to non-additive contexts) are that a functor is
For a functor between abelian categories, the last condition is equivalent to being additive and preserving short exact sequences (exercise). In any case, by default people usually only consider additive functors between abelian categories anyway.
The answer is already given by Sdevalapurkar and Qiaochu Yuan, but here is a deceptively easy proof which doesn't use category theoretic terminology:
Proposition An exact functor $F:A\operatorname{Mod}\to A\operatorname{Mod}$ sends a zero module to a zero module.
Proof The exact sequence $$0\to0\to0$$ is sent to an exact sequence $$F(0)\xrightarrow{\operatorname{id}_{F(0)}} F(0)\xrightarrow{\operatorname{id}_{F(0)}}F(0),$$ which implies that $F(0)=0$.