Understanding the slope of a line as a rate of change

I thought I was confident about what is meant by the slope of a line and how it relates to a rate of change, but I'm having doubts and I'm hoping that someone will be able to help me clear them up.

As I understand it, intuitively the slope of a line is a number $m$ that quantifies the amount it inclines from the horizontal, and its direction. Now, if $y$ is a function of $x$, such that $y=mx+c$, then one can quantify the rate of change in $y$ as one changes $x$ via the the ratio of their coordinate differences, such that $$m=\frac{\Delta y}{\Delta x}$$ Heuristically, can one (hopefully correctly) understand this quantity as follows:

The value of the function $y$ changes by an amount of $\Delta y$ units for every $\Delta x$ units change in $x$. Therefore, the value of $y$ changes by an amount $\frac{\Delta y}{\Delta x}$ units per unit change in x, which is exactly the rate of change in $y$ with respect to $x$, since it quantifies the amount the value of $y$ changes per unit change in $x$.

Would this be a correct understanding at all?

Yes, your interpretation of slope is correct. In fact, some people take this interpretation as the definition of the slope of a line.