Regardless of what your book might say, a random process is a collection of random variables that is described in mathematical notation as
$\{X(t) \colon t \in \mathbb T\}$ or $\{X_t\colon t \in \mathbb T\}$ where
$\mathbb T$ is called the index set (typically, $\mathbb T$ is
$(-\infty,\infty)$ or $\{\cdots, -2, -1, 0, 1, 2, \cdots\}$ or subsets
thereof) and we have one random variable for each $t \in \mathbb T$.
Each random variable is, of course, a mapping from the sample space
$\Omega\to \mathbb R$, and when the experiment is performed and
the outcome $\omega \in \Omega$ is known, the random variable
$X_t$ maps that $\omega$ onto some real number that is, of
course, usually denoted by $X_t(\omega)$.
Now that we have this (typically infinite) collection of random variables,
a sample path or realization of the random process is the "waveform"
that you see when the experiment is performed (with outcome
$\omega_{11809}$, say)
and each of the random
variables $X_t$ maps this outcome onto $X_t(\omega_{11809})$.
That's the "waveform": its value at $t = 0$ is $X_0(\omega_{11809})$,
and when $t = 12$, its value is $X_{12}(\omega_{11809})$ and so on.
Now we come to the crux of the matter. A deterministic random process
is one for which if we know the value(s) of one or more of the
$X_t(\omega)$, say $X_{0}(\omega)$ and $X_2(\omega)$,
then we know the value of all the $X_t(\omega)$: knowing the
value taken on by a few of the random variables in the set tells
us the values that all the random variables took on. That is,
there is "no more randomness left" in the process.
With this in mind, consider a random process for which all the
random variables $X_t$ are in fact the same random variable
$A$. (Whether $A$ is uniformly distributed on $(0,1)$ or not is
totally irrelevant to the issue being considered). Then, if we
know that $X_0(\omega) = A(\omega) = 0.13$, then
we know that $X_t(\omega) = 0.13$ for all $t \in \mathbb T$.
That is why your book is claiming that the random process
is deterministic. Note that if $X_t$ were to equal
$A \cos(2\pi f_0 t)$ where$f_0$ is some known constant instead of just $A$,
the process would still be deterministic since knowledge of the
value of $X_0$ would give away knowledge of everything.
If instead $X_t = A \sin(2\pi f_0 t)$, then knowing the value
of $X_0$ would not be useful but knowing the value of
$X_{(4f_0)^{-1}}$ (or any other $X_t$ where $\sin(2\pi f_0 t) \neq 0$)
would again be a telltale.
As a harder question that you can use to test whether you understand
the above explanation, consider whether a random process in which
by $$X_t = A \cos(2\pi f_0 t + \Theta), -\infty < t < \infty$$ where $A$ and $\Theta$
are independent random variables is a deterministic process or not.
Does knowing the values of just a few of the random variables $X_t$
tell you the value of all of them?
So, that is the basic idea behind the notion of deterministic
random processes: observation of some part of a sample path
allows us to determine the entire sample path with perfect
accuracy. How about nondeterministic random processes? Consider
$$X_n = A_n, \quad n = \cdots, -2, -1, 0, 1, 2, \cdots$$
where the $A_n$ are independent random variables with some
specific distribution, e.g. each $A_n \sim \mathcal N(0,\sigma^2)$.
In this case, the process is often referred to as a
(discrete-time) white noise process which might give you a
hint as to whether it should qualify as a deterministic or
a nondeterministic process.
Exercise: what is the significance of the number 11809 in the second
paragraph above?
