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The value of g depends on altitude. You can define it easily at the earth average 0m altitude.
It also depends on latitude, and local geology and...
Maybe it is just weighted by surface area, you're right, and that's what I meant by "surface average".
Standard gravity was adopted as a standard in 1901. That was at the 3rd meeting of the General Conference on Weights and Measures. They redefined a litre as 1 kilogram of water, but the volume of water depends on the pressure, and the pressure depends on the local gravity, so they had to come up with standard values for both standard atmosphere and standard gravity. You also need a standard value for gravity to define a standard for weight measurements which was also done.
Standard gravity is the acceleration at sea level at 45 degrees latitude. The official number was based on measurements made by Gilbert Étienne Defforges in 1888. I can't find details about his methodology without going to a library or something, and that's not worth the effort for an internet comment.
No, it's not worth it. Honestly that's great all on it's own. I guess they never had a reason to update it, then, since anybody that needs a more accurate value would just measure it themselves.
It looks like they went back to the original litre definition a few decades later. I'm not sure why they thought defining volume by mass rather than geometry was better in 1901, anyway. Some fun facts about the kilogram itself, since I never get to talk about this stuff:
Since 2019 the kilogram has been based on a "Kibble balance", which is a contraption that precisely measures the force produced by electromagnetism. The necessary electricity is provided by circuit with a material that has quantised resistance near absolute zero, and a superconducting junction which produces oscillation exactly tied to the current flowing through, which is itself timed by atomic clock. This allows you to measure it out using just the new fixed value of Plank's constant.
Before 2019 there was just a chunk of metal the was the kilogram, which is hilariously low-tech.