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 is 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 determined 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 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 at rest), 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. Clearly, the zero-input response is also a solution to the homogeneous equation, but the difference with the natural response is that the zero-input response alone satisfies the initial conditions, because it is combined with the zero-state response, which assumes zero initial conditions. On the other hand, the natural response alone does not satisfy the initial conditions. The initial conditions are satisfied only by combining the natural response with the particular solution of the difference equation (the latter being the forced response).
As mentioned above, we can write the total solution as
$$y[n]=y_{ZI}[n]+y_{ZS}[n]$$
(zero-input response plus zero-state response)
and as
$$y[n]=y_N[n]+y_F[n]$$
(natural response plus forced response). For the given example, we have
$$y_{ZI}[-1]=y[-1]$$
i.e., it is $y_{ZI}[n]$ that takes care of the initial condition. That's also why $y_{ZI}[n]=0$ if the initial condition is zero. $y_{ZI}[n]$ must satisfy the homogeneous equation
$$y_{ZI}[n]+ay_{ZI}[n-1]=0,\qquad y_{ZI}[-1]=y[-1]$$
So if $y[-1]=0$, $y_{ZI}[n]=0$ for all $n$. The natural response also satisfies the homogeneous equation, but not with the initial condition $y_N[-1]=y[-1]$. What is satisfied is $y_N[-1]+y_F[-1]=y[-1]$. This is why the natural response is generally non-zero, even for zero initial conditions. And the natural response is the homogeneous solution which we need to combine with the particular solution (forced response) found in the standard way. We usually have no direct means to find the specific particular solution which, when combined with the special homogeneous solution represented by the zero-input response, will give the complete solution of the difference equation. For this we need another homogeneous solution, and this is the natural response.
Again using the above example will hopefully clarify this. For an exponential forcing signal, the standard (and most straight-forward) way to obtain a particular solution is choosing a scaled version of the forcing function:
$$y_p[n]=Ab^n\tag{A1}$$
(for the sake of simplicity I leave out the unit step $u[n]$, assuming that we consider $n\ge 0$, unless we talk about the initial condition). The constant $A$ is determined by plugging $(A1)$ into the difference equation:
$$Ab^n+aAb^{n-1}=b^n$$
giving $A=\frac{b}{a+b}$. The general form of the homogeneous solution is
$$y_h[n]=B(-a)^n\tag{A2}$$
Of course $y_h[n]=0$ (i.e., $B=0$) is one specific solution, but that's not the one we're looking for. We need to determine the constant $B$ in such a way that the sum of the particular and the homogeneous solution satisfies the initial condition:
$$y[-1]=y_p[-1]+y_h[-1]=\frac{A}{b}-\frac{B}{a}$$
From this equation we get
$$B=\frac{a}{a+b}-ay[-1]$$
which shows that the homogeneous solution we need is non-zero if $y[-1]=0$. $y_p[n]$ and $y_h[n]$ found in this way are identical to the forced response and the natural response, respectively, as shown in $(4)$ and - implicitly - in $(2)$.
