Let X be some set and $X^r$ be the space of $r$-tuples of elements of $X$. The discrete metric on $X$ is $d(x,y) = 1$ if $x \neq y$ and $d(x,y) = 0$ if $x=y$. Now, suppose I combine discrete metrics on all of the $r$ dimensions of my tuple, using $L^1$, i.e. $d(\vec{x}, \vec{y}) = \sum_{i=1}^{d} d(x_i,y_i)$, i.e. the distance between two tuples is the number of coordinates on which they differ. This is a metric on $X^r$.
My question: Is there a commonly-used name or term for this metric? I'm sure there must be and it's just on the tip of my tongue.
Note: If $d$ had been the $L^1$ norm (i.e. $d(x,y) = \left|x-y\right|$) then the tuple metric would have been the Taxicab Metric, a.k.a. the Manhattan Distance.
This is called the $\ell_0$ distance (or $L_0$ distance), e.g., second page of these notes or this blog post.
The idea is that "the $\ell_0$ norm" of a vector is the size of its support, i.e., the number of nonzero coordinates it has. Accordingly, $\ell_0$ metric is the number of coordinates in which two points differ.
There is a terminological conflict because "$L_0$ norm" is often understood as the limit of "$L^p$ norms" as $p\to 0$, which isn't the same. So one needs to clarify what the term means.