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You are misrepresenting what Hagen von Eitzen wrote. He did *not* write that $z_1$ and $z_2$ depend on only one variable. He wrote that $f$ is a function of only one variable. That's a massive difference.

The question this refers to was whether $z_1=f'(xy)$ or $z_2=f'(x/y)$ are ambiguous, and the correct answer was that it is not because $f$ is a function of only one variable. Note that $f'(xy)$ is *not* the same as $(f(xy))'$, which would be ambiguous assuming $x$ and $y$ are both independent variables, unless we previously established a convention that e.g. primes always refer to a differentiation on $x$.

The notation $f'(xy)$ means that we take the function $f$, which takes *one* argument (let's name it $t$), take the derivative of that function with respect of its argument (no ambiguity here, there's only one argument after all), and *then* insert $xy$ as argument into the derivative we just calculated.

Note that the difference is important even in cases where only one variable is involved. For example, consider $z_3=f'(3x)$. This is *not* the derivative of $f(3x)$, the latter would be written as $z_4=(f(3x))'$ and by the chain rule would evaluate to $z_4=3f'(3x)$. Clearly $3f'(3x)\ne f'(3x)$ unless $f'(t)=0$ for all $t$.