> Fake would imply that the people releasing the population estimates have a much better estimate but are choosing to instead publish a made up number

Fake simply means not genuine. It doesn’t require the people reporting it to have a real estimate. It simply requires the people reporting it to just not try finding the real number.

How can any estimate, even a very poor estimate, be not genuine if there isn't a known better estimate? If I estimate there are 8 alien civilizations in the milky way it may be a truly terrible estimate, and the methods by which I came up with that estimate (eg one per galactic arm) may not stand up to any rigorous scrutiny, but it's as genuine an estimate as any other. To be not genuine, there must be something that is genuine, which it is not.

You don't need to necessarily know the right answer to have a fake estimate, but you have to be doing something to the estimate that you know is making it worse, which is equivalent to having the estimate where you didn't do that, which would be better.

Not even that. If I give you a fake number (by whatever definition) and you report it... the number is still fake, regardless of whether you had any inkling it might be, or whether you tried to verify it in any way.

I'm trying to think of a definition, and the best I can come up with is this: fake means the number was modified at some point without an auditable trail. For example, if I see 1 deer on a sq km and I extrapolate linearly to a 100 sq km area that there are 100 deer in that area, then the number is fake if I don't disclose the extrapolation -- and this is true even if the actual number is in fact 100 in reality.

Actually, I don't even think this covers all the bases, because it assumes there was an initially factual measurement. For example, if it that one observed deer was in fact a statue, the numbers are all fake even if everyone documented everything and acted in good faith and accidentally came up with true correct number at the end...