Why isn't the golden ratio defined as the points where $f(x)=x^3-x^2-x$ are zero? I was messing around on desmos, and when I plugged in $f(x) = x^2 - x - 1$, I get two points where $f(x)$ is zero, which are answers to the golden ratio. Why is this not used in the definition? It seems so much clearer to me.
Link: https://www.desmos.com/calculator/qmmlhbtwog

It is more clear to you, maybe. There are LOTS of equations whose solutions may be the golden ratio.
But it's definition comes from geometry, like many other mathematical constants like $\pi$ and $\sqrt{2}$.
The golden ratio is defined in this way: it's the ratio of two numbers which is also equal to the ratio between their sum and the larger of the two, that is naming $a$ and $b$ with $a >b$,
$$\frac{a+b}{a} = \frac{a}{b} = \phi$$
This definition comes in handy because it shows many interesting properties of the golden ratio such as:


*

*$\phi^{-1} = 1 - \phi$

*$ \phi^2 = \phi + 1$
It's also straightforward to derive the golden ration from this definition since it's.. the definition!
Your equation cannot be solved that easily by hands, whereas the definition for $\phi$ is immediate.
$$\frac{a+b}{a} = \frac{a}{b} = \phi$$
hence
$$\frac{a+b}{a} = 1 + \frac{b}{a} \longrightarrow 1 + \frac{1}{\phi}$$
That is
$$1 + \frac{1}{\phi} = \phi$$
That is
$$\phi^2 - \phi - 1 = 0$$
Form where the golden ration can be easily calculated.
