I’ll grant you that Cecil’s article doesn’t explicitly limit consideration to Earth’s mountains alone, but that seems understood to me.
Actually the question of the article — “Which is taller, Mt. Everest or K2?” — only obliges Cecil to address which of those two particular mountains is the taller. We can ignore all others, everywhere.
Wikipedia says it was summited in 1880. Or am I missing a joke?
Yes, you are missing a joke. Do a search on the Marianas Trench and/or Challenger Deep on these boards.
I should add that Humboldt was conducting a study of worldwide sodium chloride deposits. He was assisted by a group of nubile maidens from coastal Connecticut.
I once heard it argued that the tallest mountains in the Universe are those on the surface of neutron stars, at a couple of millimeters or so. That doesn’t seem like much, until you try to climb one: Due to a neutron star’s extreme gravity, the potential difference from base to summit dwarfs anything a mere planet can produce.
Heh. I wonder which of Earth’s mountains is highest by this definition. Probably Mauna Loa-it’s huge base-to-summit height should make up for it’s position high on the equatorial bulge …
The difference in g over the surface of the earth is pretty small, so there’s not a lot to make up for. I think the hardest part of that “climb” would be starting at the bottom. 
I’ve gotta say, OP’s like this make me gratefull for Wittgenstein, especially Philosophical Investigations. Language and meaniing are contextual, which too many armchair nitpickers seem to willfully forget.
Do we actually have evidence of such mountains (or surface irregularities)?
As for the OP - if you define the surrounding sea floor as the “base” of the mountain, then you are defining the “surface” of the earth to be where the rock (solid) ends. It’s no less arbitrary than to define “surface” as the place where the air (gas) ends.
In the interest of the Straight Dope, the equatorial bulge is 13 miles.
Yes, they figure prominently in current models explaining “glitches” in pulsar rotation. Indirect, to be sure, but then, most of what we have on neutron stars is rather indirect. Once LISA comes online, we also expect to see some gravitational-wave signature of them, which would be a bit more direct.
Another thing to think about is that Mauna Kea (or Loa) and Mt. Everest while both very large piles of rock and fundamentally different because of how they were made.
All volcanoes can be easily identified since their rocks are all slightly different from those of other volcanoes as well as the surrounding rocks. That’s how we know Kilauea and Mauna Loa are two different volcanoes coming from two sepearte sources. So if you were to make a three dimensional model of Mauna Kea you could give it a well defined base as you would know where the old sea floor stopped and where Mauna Kea began. Leaving you with several options how to measure it. Sea level to summit which is the most useful since this is the one that effects us the most. From the where the sea floor used to be to the peak. And from where the lowest rocks are that came out of the volcanoes source. I suppose you could even count the stem (as long as it’s solid) leading through the cracks in the Pacific plate from the mantle to the volcano proper although that seems to be stretching things.
Whereas Mt. Everest (which I am much less familiar with) was formed by two colossal landmases colliding and squeezing and folding the rocks of the southern edge of the Asian plate. Unlike volcanoes I don’t think you can really say where the boundary between Mt. Everest and the Asian plate exists. Really they seem to be one and the same and Everest is just a very useful word for this one extension of that plate mass. With no definable boundary I’m not sure what other ways to measure it beside sea level there are. It’s surrounded by other mountains right? I suppose you could measure from the lowest minimum around it before heading up to another mountain to its summit. I suppose you could also measure from the base of the Asian plate to its summit since there isn’t a point between them where things really change like they do for volcanoes.
P.S. I’ve been really hoping for a summit eruption on Mauna Loa to toss up a nice 120 cone. It really deserves to be taller then Mauna Kea, it’s just so massive.
P.P.S. I wonder what will happen in the Himalayas with the next big earthquake. K2 might get lucky.
All mountains are 4000 miles high?