TLDR: These formulas have nothing to do with each other. The first one is the definition of the Hadamard gate, the other one the state we're left with after having applied $f$. In the latter, the $z$ comes from the application of the Hadamard gate.
These formulas are not linked. The first one simply describes that, for a single qubit state $|x\rangle$, we have:
$$H|x\rangle=\frac{|0\rangle + (-1)^x|1\rangle}{\sqrt{2}}.$$
Indeed, we know that:
$$H|0\rangle=\frac{|0\rangle + |1\rangle}{\sqrt{2}}$$
and:
$$H|1\rangle=\frac{|0\rangle - |1\rangle}{\sqrt{2}}$$
Thus we can write:
$$H|x\rangle=\frac{|0\rangle + (-1)^x|1\rangle}{\sqrt{2}}=\frac{1}{\sqrt{2}}\left[(-1)^{x\cdot 0}|0\rangle+(-1)^{x\cdot 1}|1\rangle\right]=\frac{1}{\sqrt{2}}\sum_z(-1)^{x\cdot z}|z\rangle.$$
Note that this formula is always true, it is not specific to the Deutch-Josza algorithm, but is a property/definition of the Hadamard gate.
The second formula describes the state just after we've applied the function $f$ to the state:
$$\left|\psi_2\right\rangle=\frac{1}{\sqrt{2^n}}\sum_x(-1)^{f(x)}|x\rangle\left[\frac{|0\rangle-|1\rangle}{\sqrt{2}}\right]$$
What we want to do now is to apply an Hadamard gate on the first register. In order to do this, we can use the definition of the multi-qubit Hadamard gate which is given in the following paragraph and which can be seen as a generalization of the Hadamard gate for more than one qubit:
$$H|x\rangle=\frac{1}{\sqrt{2^n}}\sum_{z}(-1)^{x\cdot z}|z\rangle$$
This is where $z$ comes from: from the application of the Hadamard gate. Using this definition, we can now compute $\left|\psi_3\right\rangle$ by linearity:
$$\left|\psi_3\right\rangle=\frac{1}{\sqrt{2^n}}\sum_x(-1)^{f(x)}H|x\rangle\left[\frac{|0\rangle-|1\rangle}{\sqrt{2}}\right]$$
which is, according to the formula of $H|x\rangle$:
$$\left|\psi_3\right\rangle=\frac{1}{2^n}\sum_x(-1)^{f(x)}\sum_z(-1)^{x\cdot z}|z\rangle\left[\frac{|0\rangle-|1\rangle}{\sqrt{2}}\right]$$
which is also equal to:
$$\left|\psi_3\right\rangle=\frac{1}{2^n}\sum_z\sum_x(-1)^{x\cdot z+f(x)}|z\rangle\left[\frac{|0\rangle-|1\rangle}{\sqrt{2}}\right]$$
which is the state described by Nielsen and Chuang in equation (1.51).
