Monica Many very wealthy people tend not to place their money just anywhere overseas but quite often use tax havens to assist in avoiding (or evading) tax. This has been the subject of the Panorama expose, which was based on a thorough analysis of the leaked "Paradise Papers". So, I believe when people talk about "offshore investments" or "overseas investments" they are referring to these relatively recent revelations about tax havens.
As the BBC reported in July this year when discussing the issue of tax havens:
"........... Nowadays, tax havens are controversial for two reasons: tax avoidance and tax evasion.
"..........The euphemism for a tax haven these days, of course, is "offshore" ................. ..."
And an article in The Guardian, also from July this year, stated:
"The world's super-rich have taken advantage of lax tax rules to siphon off at least $21 trillion, and possibly as much as $32trillion, from their home countries and hide it abroad – a sum larger than the entire American economy.
"James Henry, a former chief economist at consultancy McKinsey and an expert on tax havens, has conducted groundbreaking new research for the Tax Justice Network campaign group – sifting through data from the Bank for International Settlements (BIS), the International Monetary Fund (IMF) and private sector analysts to construct an alarming picture that shows capital flooding out of countries across the world and disappearing into the cracks in the financial system."
You say:
" 'tax havens' are a very small subset, in every sense, of all the places where you can make overseas investments ..."
They may well be a small subset but the size of the subset doesn't necessarily determine or reflect how much money pours into it and, bearing in mind that they are now used to launder money and avoid/evade tax, the negative effect they have on the world (and especially the developing world) is likely to be disproportionate to their size. "A sum larger than the American economy" sounds like an awful lot of money to me.