You don't say what the other statistics book is, but I'd guess that it is a book (or section) about finite population sampling.
When you sample random variables, i.e. when you consider a set $X_1,\dots,X_n$ of $n$ random variables, you know that if they are independent, $f(x_1,\dots,x_n)=f(x_1)\cdots f(x_n)$, and identically distributed, in particular $E(X_i)=\mu$ and $\text{Var}(X_i)=\sigma^2$ for all $i$, then:
$$\overline{X}=\frac{\sum_i X_i}{n},\quad E(\overline{X})=\mu,\quad
\text{Var}(\overline{X})=\frac{\sigma^2}{n}$$
where $\sigma^2$ is the second central moment.
Sampling a finite population is somewhat different. If the population is of
size $N$, in sampling without replacement there are $\binom{N}{n}$ possible
samples $s_i$ of size $n$ and they are equiprobable:
$$p(s_i)=\frac{1}{\binom{N}{n}}\quad\forall i=1,\dots,\binom{N}{n}$$
For example, if $N=5$ and $n=3$, the sample space is $\{s_1,\dots,s_{10}\}$
and the possible samples are:
$$\begin{gather}s_1=\{1,2,3\},s_2=\{1,2,4\},s_3=\{1,2,5\},s_4=\{1,3,4\},s_5=\{1,3,5\},\\
s_6=\{1,4,5\},s_7=\{2,3,4\},s_8=\{2,3,5\},s_9=\{2,4,5\},s_{10}=\{3,4,5\}\end{gather}$$
If you count the number of occurrences of each individual, you can see that they are six, i.e. each individual has an equal chance of being selected (6/10). So each $s_i$ is a random sample according to the second definition. Roughly, it is not an i.i.d. random sample because individuals
are not random variables: you can consistently estimate $E[X]$ by a sample mean but will
never know its exact value, but you can know the exact population mean if $n=N$ (let me repeat: roughly.)${}^1$
Let $\mu$ be some population mean (mean height, mean income, ...). When $n<N$
you can estimate $\mu$ like in random variable sampling:
$$\overline{y}_s=\sum_{i=1}^n y_i,\quad E(\overline{y}_s)=\mu$$
but the sample mean variance is different:
$$\text{Var}(\overline{y}_s)=\frac{\tilde\sigma^2}{n}\left(1-\frac{n}{N}\right)$$
where $\tilde\sigma^2$ is the population quasi-variance:
$\frac{\sum_{i=1}^N(y_i-\overline{y})^2}{N-1}$.
Factor $(1-n/N)$ is usally called "finite population correction factor".
This is a quick example of how a (random variable) i.i.d. random sample and a
(finite population) random sample may differ. Statistical
inference is mainly about
random variable sampling, sampling
theory is about finite
population sampling.
${}^1$ Say you are manufacturing light bulbs and wish to know their average life
span. Your "population" is just a theoretical or virtual one, at least if you
keep manufacturing light bulbs. So you have to model a data generation
process and interpret a set of light bulbs as a (random variable) sample. Say
now that you find a box of 1000 light bulbs and wish to know their average
life span. You can select a small set of light bulbs (a finite population
sample), but you could select all of them. If you select a small sample, this
doesn't transform light bulbs into random variables: the random variable is
generated by you, as the choice between "all" and "a small set" is up to
you. However, when a finite population is very large (say your country
population), when choosing "all" is not viable, the second situation is better
handled as the first one.
