Why is the slope of a line defined the way it is.

Probably the formula is the way it is for historical reasons.

Generally speaking, our intuition of "steepness" matches the definition of slope: the steeper a line is, the higher is its slope in absolute value. This works with slope equal to $\text{rise} / \text{run}$, but would not work if the slope were equal to $\text{run} / \text{rise}$ as you suggest.

For a modern example, when you drive down a steep mountain highway, you will notice signs saying something like "Danger: 5% grade ahead. Trucks shift to low gear". This sign is informing the truck driver that the slope, as a ratio of rise over run, is equal to $-.05$ (which, to a truck driver, is dangerously steep). A higher percentage represents a higher absolute value of slope represents a steeper highway represents more danger for the truck driver.

It doesn't really matter whether we consider "change in $y$ over change in $x$", or the reciprocal - can mix up our definitions as long as we mix up the way we think about things, in the exact same way. In the reciprocal version, slopes closer to $0$ would be more nearly verical, rather than slopes nearer $\pm\infty$, as they are now.

That said, with $y$ typically being the dependent, or response variable, we have a good precedent for making it the top of the fraction. We measure automobile efficiency in miles per gallon, where miles are really the response variable. We calculate normalized costs as price per unit; it's more natural to think of the unit, the denominator, as stable; it doesn't depend on anything. Psychologically, that's just the way humans (at least in my neck of the woods) like things, in all the examples I can think of.

When we define slope, we first have to figure out what we mean by slope. Usually we want the slope to be a number which represents the "steepnes" of the line, and a more steep line should have a larger slope.

The way we defined the slope exactly is: The amount that $y$ changes by, when we change $x$ by $1$. This is just a definition we could have said "The amount that $y$ changes by, when we change $x$ by $2$." instead, however we like simplicity and $1$ is the simplest number to use here.

The above definition gives the formula you know: $$ \frac{y_2-y_1}{x_2-x_1} $$

As for the similar formula $$ \frac{x_2-x_1}{y_2-y_1} $$ This actually doesn't serve us well as slope, because the steeper the line would be, the smaller the slope. That's the opposite of what we wanted.

The intuitive concept of ''slope'' can become a rigorous mathematical concept only if we define some straight line to be ''horizontal'' and some other to be ''vertical''. If you draw a line on a paper, and you does not fix a coordinate system, this line has no well defined slope. Only wen you define two orthogonal lines as axis of an orthogonal reference system, your intuitive concept can be well defined. Usually we represents the horizontal axis as $x$ axis and the vertical as $y$ axis ( this is matter of convention), so the slope, can be defined as the ratio between the vertical displacement and the corresponding horizontal displacement, i.e. $\dfrac{y_2-y_1}{x_2-x_1}$.

As a consequence: in a non orthogonal coordinate system, the concept of slope become less interesting.