calculus - Meaning behind differentials

So I think I understand what differentials are, but let me know if I'm wrong.

So let's take $y=f(x)$ such that $f: [a,b] \subset \Bbb R \to \Bbb R$. Instead of defining the derivative of $f$ in terms of the differentials $\text{dy}$ and $\text{dx}$, we take the derivative $f'(x)$ as our "primitive". Then to define the differentials we do as follows:

We find some $x_0 \in [a,b]$ where there is some neighborhood of $x_0$, $N(x_0)$, such that all $f(x)$ in $\{f(x) \in \Bbb R \mid x \in N(x_0)\}$ are differentiable. Then we choose another point in $N(x_0)$, let's call it $x_1$, such that $x_1 \ne x_0$. Then let $dx = \Delta x = x_1 - x_0$. Now this $\Delta x$ doesn't actually have to be very small like we're taught in Calculus 1 (in particular it's not infinitesimal, it's finite). In fact, as long as $f(x)$ is differentiable for all $x \in [-10^{10}, 10^{10}]$ we could choose $x_0 = -10^{10}$ and $x_1 = 10^{10}$.

Then we know that $\Delta y = f'(x_0) \Delta x + \epsilon(\Delta x)$, where $\epsilon(\Delta x)$ is some nonlinear function of $\Delta x$. If $f(x)$ is smooth, we know that $\epsilon(\Delta x)$ is equal to the sum of powers of $\Delta x$ with some coefficients, by Taylor's theorem. But of course, $\epsilon(\Delta x)$ won't be so easy to describe if $f(x)$ is only once differentiable. So we define $dy$ as $dy = f'(x_0) dx$: that is, $dy$ is the linear part of $\Delta y$. This has the very useful property that $\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \frac{dy}{dx} = f'(x_0)$. This is then not a definition of the derivative, but a consequence of our definitions.

It can be seen from this $dy$ really depends on what we choose as $dx$, but $f'$ is independent of both.

This definition can be extended to functions of multiple variables, like $z = f(x, y)$ as well, by letting $\Delta x = dx,\ \Delta y=dy$ and defining $dz$ as $dz = \frac{\partial f(x_0, y_0)}{\partial x}dx + \frac{\partial f(x_0, y_0)}{\partial y} dy$. So $dz$ is the linear part of $\Delta z$. Does all of the above look correct?

If so, then where I'm having a problem is:
1) how then do we define the derivative of $f(x)$ if not by $f'(x_0) = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$?
2) how do we apply this definition of $dx$ to $\int_a^b f(x)dx$? It seems like the inherit arbitrariness of $dx$ is really going to get in the way of a good definition of the integral.