elementary number theory - Why is this proof that $sqrt{2}$ is irrational titled as "Proof by infinite descent"?

I am reading this wikipedea article on the proof of irrationality of $\sqrt{2}$. It uses the principle of infinite descent. I understand it as:

1. We assume $\sqrt{2}=\dfrac pq$, where $p$ and $q$ are some positive integers.


2. We have $2q^2 = p^2 \implies$ $p$ is even, i.e. $p=2r$ for some positive integer $r$.


3. Now we have $2q^2=(2r)^2 \implies q$ is also even, i.e. $q=2s$ for some positive integer $s$.

From steps 1,2 and 3 we conclude that both $p$ and $q$ have $2$ as their factor at least one time. So we can say $\sqrt{2}=\dfrac pq = \dfrac rs$.

Now we can repeat the steps 1,2 and 3 with $\sqrt{2}= \dfrac rs$. This will eventually imply that intezers $r$ and $s$ also have 2 as their factor at least one time, or $r=2r_1$ and $s=2s_2$. Recalling $p=2r$ and $q=2s$ implies that $p$ and $q$ have $2$ as their factor at least two number of times.

The notable fact is that we can repeat steps 1,2 and 3 infinite number times, which implies that $p$ and $q$ have 2 as their factor infinite number of times, but this can't be for any finite $p$ and $q$, that is to say their does not exist any intezer which can have 2 as its factor infinite number of times. So we conclude that there doesn't exist any $p$ and $q$ which satisfies $\sqrt 2 = \dfrac pq$, hence $\sqrt 2$ is irrational.

Question 1: Do I understand the proof correctly?

Question 2: I noted that this proof on wikipedea is titled as Proof by infinite descent, where it does not use the concept of infinite descent at all. So why is it titled as Proof by infinite descent?