For the record, the handheld VHF I was using to issue the position report was in my TZT14 buddy list. Also, the hexagon icon and entry in the DSC list both disappear if the TZT14 is power-cycled.
When I first enabled the DSC feature in my fixed VHF radio and found I couldn't delete the individual position reports displayed on the TZT14, I became concerned that the screen might become filled with them during a day of cruising. I ran another test and found that each position report from the same MMSI replaces the previous, so there is only one hexagon (and one DSC list entry) for each MMSI. That plus the fact that there will be only 20 hexagons at any one time is enough to eliminate my concerns.
---rick