I have not seen concretely your metric, but something similar is used to analyze extrema of Gaussian random fields. A very useful notion is entropy with respect to the following metric: $d(s,t)=(\mathbf{E}(X_t-X_s)^2)^{1/2}$. I think, such metrics were introduced by Dudley, but I may be mistaken. I am sure this is written in many books and papers. A minute of googling gave me http://cims.nyu.edu/~zeitouni/notesGauss.pdf, where this metric is introduced on p.18.

Your independence assumption makes this metric not so useful though, it seems, and the following computation may be more elucidating for your problem:

$$
E|X-Y|= E\int_R (1_{X\le x}-1_{Y\le x})^2 dx = \int_R E(1_{X\le x}+1_{Y\le x}-21_{X\le x}1_{Y\le x})dx
=\int_R (P\{X\le x\}+P\{Y\le x\}-2P\{X\le x\}P\{Y\le x\})dx
=\int_R ((P\{X\le x\}-P^2\{X\le x\})+(P\{Y\le x\}-P^2\{Y\le x\})+(P\{X\le x\}-P\{Y\le x\})^2)dx
=\int_R F_X(x)(1-F_X(x)) dx +\int_R F_Y(x)(1-F_Y(x)) dx + \int_R(F_X(x)-F_Y(x))^2dx
$$

The first term on the right-hand side depends only on $F_X$, the distribution of $X$, the second one depends only on $F_Y$, the distribution of $Y$,
and the last one is the square of $L^2$ distance between the distribution functions of $X$ and $Y$.

So the distance you want to consider is, in fact, a kind of $L^2$ distance on distributions with a little bit of what is called the "nugget
effect" in geostatistics, where they often consider random fields $X_t$ such that $E(X_s-X_t)^2$ does not converge to zero as $s\to t$. This
occurs naturally if $X_t=Y_t+Z_t$, where $Y_t$ is a very nice smooth process, and $Z_t$ has zero correlation range, i.e., $cov(Z_t,Z_s)=0$ for $t\ne s$.