The article is likely referring to the estimated dry prominence of Mauna Kea. Topographic prominence [1] measures the height of a peak relative to its lowest surrounding contour line (that doesn't also contain another, taller peak). Usually the ocean's surface is considered "flat ground" when computing this measure (called a "wet prominence"), but one can alternatively imagine the Earth to have no water, and compute a "dry prominence" that permits the lowest surrounding contour line to be below sea level.

[1]: https://en.wikipedia.org/wiki/Topographic_prominence

The above Wikipedia article specifically mentions the dry prominence of Mauna Kea, estimating it to be 9330m which is taller than Everest's conventional height of 8848m. However, an apples-to-apples comparison would use the dry prominence of Everest too, which is the distance from the bottom of the Challenger Deep (-10,911m) to the summit. This would make Mauna Kea the second tallest mountain by topographic prominence if the Earth had no oceans.

Yeah that has always seemed a bit jumbled to me. It's not clear to me why that point, 33,500 feet below the peak of Mauna Kea, is appropriate to use as a basis of measurement for the mountain. And why not a lower point? And why can that point not be used to measure the height of Mount Everest as well?

Anyway, I think what you're asking can usually be described by the concept of Topographic Prominence[1], one definition of which is "the height of the peak’s summit above the lowest contour line encircling it but containing no higher summit"

[1] https://en.wikipedia.org/wiki/Topographic_prominence

It's wet :) (under water surface)

The claim of the article that mount Everest is the tallest point from the earth's center happens because earth's surface, relative to its size, is as smooth as a billiard ball. Even the tallest mountains are nothing. So high mountains located at the equator can easily "outperform" higher mountains elsewhere.

When comparing the actual height of a mountain, they're measured from their base (which is whatever elevation the ground they sit on is) to their peak, rather than just looking at their peak altitude. This is how you compare mountains on other planets, and why Mons Olympus on Mars is the tallest mountain in the solar system. For Mauna Kea, the mountain is taller than it looks because its base sits at the bottom of the ocean, whereas Everest sits on an already-lofty piece of continental plate.

Right, basically, in a different context accrding to this principle volcanoes at higher altitudes (say close to andes but not in the andes proper) are not as tall as more coastal/lower altitude volcanoes, if the cone’s rise isn't as high.