They are waves, but they don't behave like the sort of waves we are used to. This is the source of all the confusion.

I have heard description of a tsunami being "a temporary rise in sea level", which describes its behavior much more intuitively. A tsunami that tops a sea wall will flood the entire lower-lying area behind it. A usual wave, even a tall one, will only deposit some splashes of water behind the wall and go away immediately.