Ok, so I think my initial reaction and the obvious answer is
correct:

Positive weights **cannot** affect the **shortest
path** between two nodes in a binary tree.

On the other hand, weights obviously **do** affect the
**shortest 'distance'** between two nodes in a binary tree as
compared to simply calculating 'distance' between nodes as the number of
hops.

This is what the udacity instruction was getting at.
It seems that this instruction to work with weighted binary trees was
simply to enable correct automatic grading of the code which relied on
using the labels to calculate the exact shortest
'**distance**' (which is affected by weight) as opposed to the
shortest **path** (list of nodes) which is not.

Once I modified my algorithm to take this into account and output the
correct distance, it passed the grader.