How is $\lambda x.x^2$ a valid lambda expression?

According to wikipedia:

• a variable, $x$, is itself a valid lambda term
• if $t$ is a lambda term, and $x$ is a variable, then $(λ x . t )$ is a lambda term (called a lambda abstraction);
• if $t$ and $s$ are lambda terms, then ( $t$ $s$ ) ($ts$) is a lambda term (called an application).

How can $\lambda x.x^2$ be a valid lambda term if $x^2$ is not, as it is not a variable and is neither an abstraction nor an application?

I would interpret it as $$\lambda x.\ \mathrm{m}\ x\ x$$ where $\mathrm{m}$ stands for the multiplication operation in context (or $\lambda x.\ \mathrm{p}\ x\ 2$ where $\mathrm{p}$ is the power opertion and $2$ stands for the term corresponding to numeral $2$).
Please note, that we use syntactic sugar often and pure lambda calculus is rather rare, e.g., $$\mathrm{let}\ i = \lambda x.\ x\ \mathtt{in}\ i\ \ i$$ and $$i\ i\ \mathrm{where}\ i = \lambda x.\ x$$ both actually stand for $$(\lambda i.\ i\ i)\ (\lambda x.\ x)$$ which is a way harder to read. It is quite inconvienient to use just pure lambda calculus.
I hope this helps $\ddot\smile$