I learnt we have different definitions for the total variation for functions of the form $f:\mathbb{R}^2\to\mathbb{R}$ which are in some way analogous to the total variation of functions of one variable.
For $y = f(x)$, where $f$ is smooth, the total variation is equal to the arc length of the curve $(x,f(x))$. Is there any such characterization for functions of two variables, such that the total variation is nothing but the surface area of $f(x,y)$, when $f$ is smooth?
Your description in one dimension isn't quite right. The length of the curve $y = f(x)$ on the interval $[a,b]$ is $\displaystyle \int_a^b \sqrt{1 + |f'(x)|^2} \, dx$, but the total variation of $f$ is just $\displaystyle \int_a^b |f'(x)| \, dx$.
In two dimensions, the total variation of a smooth function $f$ on a nice open set $\Omega$ (say a disk or a rectangle) is $\displaystyle \iint_\Omega |Df(x)| \, dx.$
The definition of variation of nonsmooth functions is quite a bit more sophisticated in two dimensions than in one.