I think they need to call it what it is: a tsunami. Storm surge makes it sound like it's something you can run away from.
The thread you made is outstanding. Very few people know how decimating the storm surge can be.
A storm surge will be there as long as the storm keeps stacking the water up ahead of it, and the wind and tides can combine to increase the devastating effect.
You are right about the level of devastation, the effect is pretty much the same, with the added destructive force of the wind whipping waves into everything. That constant, rhythmic battering is more destructive in the long term than just an influx of water.
Over all, for a given depth, the storm surge is worse except in the regard that it can be predicted and avoided with a greater span of time to escape. Property suffers. People need not, except for the loss of what they cannot take with them.
If you want to call it what it was, then you wouldn't call it a tsunami.