Question: In the data structure field, what is the (best) term used to identify (designate) a subtree covering a subset of contiguous leaves?

Answer: 
You seem to mean a subtree obtained by taking a certain node and all its successors.

In data structures, the implicit assumption usually is that we consider *rooted* (and often *ordered*) trees. Therefore, "subtree" is usually taken to mean what you intend; "subtree" as in graph-theoretic "subgraph" is rarely used (in my experience).

If you want to make very clear what you mean, you can write

> 
> The subtree of $T$ which is rooted in $x \in T$.
> 
> 
> 

A common notation for this tree is $T\_x$.

[![enter image description here](https://i.stack.imgur.com/QFYbR.png)](https://i.stack.imgur.com/QFYbR.png)
It is the subtree rooted at the *least common ancestor* of the set of leaves (the fact that the leaves are continuous makes no difference). I'm not sure there is any shorter term.