Isn't my book using the summation notation incorrectly when writing $\lim_{\Delta x\to0}\sum_{n=1}^{N}f(x)\Delta x$?

That is indeed sloppy notation. The proper way of writing it is to say the following. Suppose $f:[a,b]\to\Bbb{R}$ is a given bounded function. Let $P=\{x_0,\dots, x_N\}$ be a partition of the interval $[a,b]$, meaning that $a=x_0<x_1<\cdots< x_N=b$. Suppose also that we're given a collection of points $\{\xi_1,\dots, \xi_N\}$, where $\xi_1\in[x_0,x_1], \cdots, \xi_N\in [x_{N-1},x_N]$. We call $(P,\{\xi_i\}_{i=1}^N)$ a tagged partition of $[a,b]$.

Corresponding to this tagged partition, we consider the Riemann sum \begin{align} R(f,P,\{\xi_i\}_{i=1}^N):=\sum_{i=1}^Nf(\xi_i)(x_i-x_{i-1})\equiv\sum_{i=1}^Nf(\xi_i)\Delta x_i \end{align} where we defined $\Delta x_i:=x_i-x_{i-1}$ for each $i\in\{1,\dots, N\}$. Here, $\xi_i$ is just an intermediate point of the interval $[x_{i-1},x_i]$. Also, let us define $\|P\|:=\max\limits_{1\leq i\leq N}\Delta x_i$; this is called the mesh of the partition $P$.

Finally, with all the notation above, we say that the function $f$ is Riemann integrable on $[a,b]$ if the following limit exists: \begin{align} \lim_{\|P\|\to 0}R(f,P,\{\xi_i\}_{i=1}^N) \end{align} i.e we require the limit \begin{align} \lim_{\|P\|\to 0}\sum_{i=1}^Nf(\xi_i)\Delta x_i\tag{$*$} \end{align} to exist. More explicitly, what this means is:

There is a number $I\in\Bbb{R}$ such that for every $\epsilon>0$, there is a $\delta>0$ such that for any tagged partition $(P=\{x_i\}_{i=0}^N,\{\xi_i\}_{i=1}^N)$ of $[a,b]$, if $\|P\|<\delta$ then \begin{align} \left|R(f,P,\{\xi_i\}_{i=1}^N)-I\right|:=\left|\sum_{i=1}^Nf(\xi_i)\Delta x_i - I\right|<\epsilon. \end{align}

One can easily show that if the number $I$ exists, then it is unique.

In this case, we call the unique number $I$ the Riemann integral of $f$ on the interval $[a,b]$, and write this as $\int_a^bf$. We define this to be the (signed) area bounded by the graph of $f$ on the interval $[a,b]$ and the horizontal axis.

So long story short, $(*)$ is what you should be writing. The choice of index $i$ is irrelevant. I can use $j$ or $\alpha$ or $n$ or any other letter I like, for example, $\sum_{\beta=1}^Nf(\xi_{\beta})\Delta x_{\beta}$. Writing $\sum_{n=1}^Nf(x)\Delta x$ is incorrect for two reasons. First, it makes it seem like $\Delta x$ is a fixed number and $x$ is fixed, so the sum should just evaluate to $Nf(x)\Delta x$; of course this isn't the intended meaning.

We have to allow several things to vary: the number of points in the partition (i.e $N$), and also the spacing (i.e we're not assuming all the $\Delta x_i$'s are equal), and also we have allow for arbitrary intermediate points (i.e arbitrary choice of $\xi_i$'s).