Can we say that $\sqrt{2}=2/(2/(2/(\dots )))$?

We have

$\varphi =1+\frac{{\textstyle 1}}{{\textstyle \varphi}}=1+\frac{{\textstyle 1}}{{\textstyle 1+\frac{{\textstyle 1}}{{\textstyle \varphi}}}}=1+\frac{{\textstyle 1}}{{\textstyle 1+\frac{{\textstyle 1}}{{\textstyle 1+\frac{1}{\varphi}}}}}=\cdots $

(with $\varphi $ being the Golden Ratio)

Which gives us the confirmed infinite fraction

$\varphi =1+\frac{{\textstyle 1}}{{\textstyle 1+\frac{{\textstyle 1}}{{\textstyle 1+\frac{{\textstyle 1}}{{\textstyle \ddots}}}}}}$

We also have

$\sqrt{2}=\frac{{\textstyle 2}}{{\textstyle \sqrt{2}}}=\frac{{\textstyle 2}}{{\textstyle \frac{{\textstyle 2}}{{\textstyle \sqrt{2}}}}}=\frac{{\textstyle 2}}{{\textstyle \frac{{\textstyle 2}}{{\textstyle \frac{{\textstyle 2}}{{\textstyle \sqrt{2}}}}}}}=\cdots $

So by analogy we can deduce that

$\sqrt{2}=\frac{{\textstyle 2}}{{\textstyle \frac{{\textstyle 2}}{{\textstyle \frac{{\textstyle 2}}{{\textstyle \frac{{\textstyle 2}}{{\textstyle \vdots}}}}}}}}$

The sequence $({a}_{n}{)}_{n\in {\mathbb{Z}}^{+}}$ such that ${a}_{1}=\sqrt{2},{a}_{n+1}=\frac{2}{{a}_{n}}$ gives $\underset{n\to +\mathrm{\infty}}{lim}{a}_{n}=\sqrt{2}$, so indeed the representation should be correct.

Everywhere on the Internet that I see a continued fraction of $\sqrt{2}$, it is

$\sqrt{2}=1+(\sqrt{2}-1)=1+\frac{{\textstyle 1}}{{\textstyle 1+\sqrt{2}}}=1+\frac{{\textstyle 1}}{{\textstyle 2+\frac{{\textstyle 1}}{{\textstyle 1+\sqrt{2}}}}}=1+\frac{{\textstyle 1}}{{\textstyle 2+\frac{{\textstyle 1}}{{\textstyle 2+\frac{{\textstyle 1}}{{\textstyle 1+\sqrt{2}}}}}}}=\cdots $

Why haven't I seen the representation $\sqrt{2}=2/(2/(2/(2/\dots )))$ mentioned, and see the above used instead? Is something wrong about my representation?

I would say it isn't mentioned because it is not useful: you cannot approximate $\sqrt{2}$ using the representation, unlike you can using the above one (which you can do by removing $\frac{1}{1+\sqrt{2}}$ n any member).