What do you call a subtree covering a subset of contiguous leaves?
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 $T_x$.