Explanation for the reactions in a saltwater battery with zinc and copper electrodes
This answer deals with the situation, when but electrodes are in the same salt water solution.



Assuming your salt solution contains sodium chloride
$$\ce{NaCl (s) -&gt;[\ce{H2O}] Na+ (aq) + Cl- (aq)},$$
then your reaction at the zinc terminal will certainly be
$$\ce{Zn (s) + 2Cl- -&gt; ZnCl2 (aq) + 2e-}.$$
On the copper terminal water will be electrolysed
$$\ce{H2O + 2e- -&gt;[\ce{Cu}] 1/2 H2 ^ + {}^{-}OH},$$
or using autoprotolysis
\begin{aligned}\ce{
2H2O &amp;&lt;=&gt; H3+O + {}^{-}OH\\
H3+O + e- &amp;-&gt;[\ce{Cu}] 1/2 H2 ^ + H2O
}.\end{aligned}

This will happen since sodium has a much lower electron affinity than water/ hydronium
\begin{aligned}
\ce{Na+ + e- &amp;-&gt; Na} &amp;E^\circ&amp;=−2.71\:\mathrm{V}\\
\ce{2H2O + 2e- &amp;-&gt; H2 + 2 {}^{-}OH}&amp;E^\circ&amp;=−0.83\:\mathrm{V}\\
\ce{2H+ + 2e- &amp;-&gt; H2}&amp;E^\circ&amp;=0\:\mathrm{V}.
\end{aligned}

Depending on what salt solution you are using, different cations might be reduced. You can find a collection of values at wikipedia.



In a battery (Galvanic cell) there are usually coupled half cells, and in this case this would probably be $\ce{Zn|Zn^2+||Cu^2+|Cu}$.

This means, that the Zinc terminal is in a zinc containing solution and the copper terminal is in a copper containing solution. In this case your assumption is correct 
\begin{aligned}
 \text{Zinc terminal:}&amp;&amp;\ce{Zn(s) + ZnCl2 (aq)&amp;&lt;=&gt; 2Zn^{2+}(aq) + 2Cl- + 2e^-} \\
 \text{Copper Terminal:}&amp;&amp;\ce{CuCl2(aq) + Cu (s) + 2e^- &amp;&lt;=&gt; 2Cu(s) + 2Cl-} 
\end{aligned}
There has to be a salt bridge to neutralise charge differences (ion flow).
Note that in that case zinc ions will go into solution, while copper ions will be removed from the solution. If the concentration of copper drops close to zero, then no current can be measured. The galvanic cell can be charged by "reversing" the process, apply opposite current. Then zinc ions will be removed from solution and copper will go into solution.
In reality, this is a equilibrium process.