Above ground estimates always include all forms of gold (bullion or otherwise), so there should be no discrepancies there.

As for the gold hoard, the numbers cited are usually unrealistic (tens of thousands of tons). They are unrealistic mainly because gold production was extremely inefficient in the past even in the richest sites. Today we are close to 3000tons per year with very advanced means (compared to old times or ancient times) and still it'd take us 30-40 years to mine 100k tons. From the entire globe nonetheless...

I've also heard the story of Karen Hudes. I don't know what to make of it. The numbers don't add up in terms of production and extrapolating such levels of production in the past...

In any case the billionaire says 300k tons and that's a huge deviation from the 180k. Who knows what's going on... but old gold doesn't explain it well for me. I'd be more satisfied with an explanation like "we are mining 6k tons per year, 3k is acquired by miners to the Elite without being declared anywhere, so over 30-40 years they have +100k tons" - or something to that effect. This would make more sense for me.