Note that the three equations are different only in whether the nickel and cadmium oxides are hydrated (and thus where water appears in the equation). Also, the Chemwiki equation is not balanced because nickel oxide is $\ce{Ni2O3}$ and not $\ce{NiO2}$.
The correct equation for the ChemWiki version is:
$$\ce{Ni2O3 + Cd + 3H2O -> 2Ni(OH)2 + Cd(OH)2}$$
Many metal oxides react with water to form hydroxides. This kind of behavior is probably most frequently taught with the alkali metals and alkaline earth metals:
$$\begin{align}
\ce{2NaO + H2O} &\ce{-> 2NaOH}\\
\ce{CaO + H2O} &\ce{-> Ca(OH)2}\end{align}
$$
However, other metals undergo these kinds of reactions. For example, the chemistry of rust involves the equilibrium between iron oxides and iron oxide hydroxides:
$$\begin{align}
\ce{FeO + H2O} &\ce{<=> Fe(OH)2}\\
\ce{Fe2O3 + 3H2O} &\ce{<=> 2Fe(OH)3}\end{align}
$$
Nickel
The three ionic nickel species in the reactants are different only on the degree of hydration:
$$\begin{align}
\ce{Ni2O3 + H2O} &\ce{<=> 2NiO(OH)}\\
\ce{NiO(OH) + H2O} &\ce{<=> Ni(OH)3}\end{align}
$$
Cadium
The two ionic cadmium species in the products differ only by degree of hydration:
$$
\ce{CdO2 + H2O<=> Cd(OH)2}
$$
