Yes, Canie. I agree that both your cited sites do use that term. But I suspect they are using the term casually (informally) or deliberately avoiding confusing a lay public which may not comprehend the term "run-up heights." After all, it is unlikely anyone was there at the time with the correct instruments, or excellent observational skills, along with the necessary sangfroid to measure the actual height of the wave top.

It may be the point is moot as an expert may weigh in with the opinion that both measurements are one and the same. But I think they aren't. Momentum would carry the wave to additional height. And there's a LOT of momentum.

MW