Variation to edit distance depending on position still a metric?
Here is why the usual edit distance is a metric:


$d(w,w) = 0$ since no operations need to be performed to get from $w$ to $w$.
$d(x,y) = d(y,x)$ since given a sequence of operations for transforming $x$ to $y$, we can perform it in reverse to transform $y$ to $x$ at the same cost. This shows that $d(y,x) \leq d(x,y)$, and similarly $d(x,y) \leq d(y,x)$.
$d(x,y) \leq d(x,z) + d(z,y)$ since if $\alpha$ transforms $x$ to $z$ and $\beta$ transforms $z$ to $y$ then $\alpha\beta$ transforms $x$ to $y$, and moreover the cost of $\alpha\beta$ is the sum of the costs of $\alpha$ and $\beta$.


Now you're in a position to answer your question yourself.