First it's important to realize that many authors use the terms zero-input response and natural response as synonyms. This convention is used in the corresponding wikipedia article, and for instance also in this book. Even Proakis and Manolakis are not entirely clear about it. In the book you quoted you can find the following sentence on page 97:
[...] the output of the system with zero input is called the zero-input response or natural response.
This suggests that the two terms can be used interchangeably. Further down the page, we find the following sentence:
Thus the zero-input response is a characteristic of the system itself, and it is also known as the natural or free response of the svstem.
Again, this strongly suggests that the authors believe that both terms are equivalent.
However, on the pages you mentioned they appear to make a difference between the two. And the difference is as follows. The zero-input response is the response which is caused by non-zero initial conditions. It only depends on the system properties and on the values of the initial conditions. The zero-input response becomes zero if the initial conditions are zero.
The natural response is the part of the total response the shape of which is only determined by the poles of the system, and which doesn't depend on the poles of the (transform of the) input signal. The natural response does depend on the input signal in terms of constants but its form is entirely given by the system's poles. Unlike the zero-input response, the natural response does not vanish for zero initial conditions.
The total response of the system can be written as the following two sums:
- zero-input response + zero-state response
- natural response + forced response
The zero-state response is the response for zero initial conditions, and the forced response is the part of the response the form of which is determined by the form of the input signal.
I hope this becomes clear in the following example. Let's investigate the following system:
$$y[n]+ay[n-1]=b^nu[n],\qquad y[-1]=c\tag{1}$$
where $u[n]$ is the unit step sequence. The total response can be computed using $\mathcal{Z}$-transform techniques:
$$y[n]=\left[\frac{1}{a+b}b^{n+1}+\left(c-\frac{1}{a+b}\right)(-a)^{n+1}\right]u[n]\tag{2}$$
The zero-input response is the part of the total response that is determined by the initial condition and that does not depend on $b$:
$$y_{ZI}[n]=c(-a)^{n+1}u[n]\tag{3}$$
Obviously, $y_{ZI}[n]=0$ for $c=y[-1]=0$, i.e., for zero initial condition.
The natural response is the part of the total response the shape of which is determined by the system's pole:
$$y_N[n]=\left(c-\frac{1}{a+b}\right)(-a)^{n+1}u[n]\tag{4}$$
Note that it depends on the initial conditions as well as on the input signal (via the constant $b$).
Also note that it is the shape of the zero-state response that depends on the poles of the system as well as on the poles of the input signal transform. All other responses mentioned here only depend on one of the two sets of poles. The shapes of of the zero-input response and of the natural response depend only on the system's poles, whereas the shape of the forced response is determined by the poles of the input signal. The expression for $y[n]$ quoted in your question from Proakis and Manolakis is the zero-state response (because the system is initially relaxed), and the first sum is the forced response, and the second sum is the natural response. Since the zero-input response is zero in this case, the sum of natural response and forced response (i.e., the total response) equals the zero-state response
In mathematical terms, the natural response is the homogeneous solution of the difference equation, where the constants are determined such that the sum of the particular solution (the forced response) and the homogeneous solution satisfy the given initial condition.