It's a distinction without value I think. There are waves, and many of them. There is a rise in the sea level. For anywhere affected, both certainly matter. Like you mentioned, tsunami isn't a brief event. And here in Japan, they are talking about tsunami waves, not a singular tsunami. And talking about sea level rise and checking the local power poles for sea level indicators from previous tsunami events and floods.
I is absolutely a VERY valuable distinction because the behavior as it affects humans (up to and including killing them) is VERY different.
Regular waves that are a little higher than your seawall might cause some water damage to the buildings right next to it. A tsunami that is a little higher than your seawall will flood your entire town and drown people who are caught in basements.
Sure, but if you insist it's like a tide you downplay the risk of the initial hit of the wavefronts and the potential for it to slam up the coast or a seawall becoming a larger local wave. And if you insist it's like a wave, you downplay the persistent risk of both follow-up waves and ongoing flooding that won't subside quickly.
So saying it's not waves is dangerous, and saying it's not a sea level rise is dangerous. It's not useful to try and delineate between a tsunami being one of the two when it's in reality an event that consists of both.
(Ignoring that a sea level rise and a long-wavelength wave are the same thing)