Mount Everest - how tall (from base)?

I would not say they are "wrong". They just measure against a different baseline elevation than you want.

What is the "base" for a mountain? That is usually not really a definable term, but as a general idea, one can think of a prominence such as a mountain as rising above a surrounding plain (a flatland even if not truly flat), and then use the elevation of that plain as the baseline. There is a lot of uncertainty involved in the definition of the elevation of a plain such as that except if the plain is truly planar, which would be very unusual.

Clearly, the Himalayas rise above surrounding "plains" whether to the north (China and Tibetan Plateau) or to the south (the Indian subcontinent), but those are at different elevations so even if you use that concept of a base being what the land would look like if the surrounding non-mountainous land continued at the same general pattern or slope, you would end up with different base elevations depending on which way you do the extrapolation.