Short answer: All genes are present in both B and T cells, but they rearrange and express receptors that are specific to their lineage. Their are independent VDJ regions for B cell receptor generation and T cell receptor generation! This wiki article sheds light on this:
I too tragically missed this.. but it is actually so simple that we are missing the forest through the trees --
While both B and T cells undergo VDJ rearrangement, the V, D, and J genes are DIFFERENT GENES despite being organized and therefore classified in a similar fashion due to their shared need to generate diverse structures. To make matters more confusing, both sets of VDJ genes are on chromosome 14.
This is why the NBME specified that they are tagging THE SAME J REGION which is that of the B cell receptor gene J region, not the T cell receptor gene J-beta region (see wiki article). The B cell receptor gene is not rearranged in a T cell, if it was, T cells would have antibodies!