Money is fungible. And things get lost in translation, especially between micro and macro.
"Helicopter money" is when the central bank prints money, gives it to the government, and the government gives it to everyone, as a freebie.
When is helicopter money optimal?
Preliminary answer: almost all the time. Because almost every year, it is optimal for the Bank of Canada to print money to keep inflation at 2%. And it earns seigniorage profits from printing money, and it gives those profits to the government. And it is optimal for the government to give money to some people, like the very poor. And you could say that those welfare cheques are financed (in part) by printing money. Or you could say that it is road repairs that are financed by printing money. Since money is fungible, you could say that anything (but not everything) the government spends money on is financed by printing money.
But welfare isn't universal; the helicopter drops the money only on the poor, unlike a real helicopter. And I'm playing fast and loose with fungibility. So you might not like this answer.
Better answer: For simplicity, assume a linear tax/transfer system so that the amount of tax Ti paid by individual i depends on his income Yi: Ti = A+bYi. The parameter 'A' could well be a negative number, like under the Negative Income Tax system. (We would still have a constant term for any non-linear tax system, so I don't really need that assumption.) If A is negative, you can think of it as a lump-sum transfer payment that every individual gets from the government, and if b=0.5, they then pay 50% tax on all their income.
There is a whole micro "Optimal Tax" literature that tells you how to set the parameters A and b. Under any vaguely reasonable Social Welfare Function and utility function, the optimal A will be negative for a rich country (we don't let anyone starve). What does (should) A depend on? Ummmm, a whole host of stuff. Like: GDP per capita (richer countries should have a more negative A); the marginal benefit curve for government spending on goods (a rightward shift in that curve should make A less negative, because it means higher spending and taxes are optimal); a load of distributional and consumption/leisure preference stuff (ask Frances); and in an intertemporal model it should also depend on a load of expectations about the future and on the real interest rate (a lower real interest rate should mean a more negative A this period relative to future periods, because a lower real interest rate means you want to do nice things now and postpone nasty things into the future).
Unless none of those things ever change over time, the optimal universal lump sum transfer payment -A will change from year to year. It will change partly in an anticipated way, and partly in response to new information, or "news". In some years the government will get news which tells it to increase that transfer payment.
Now let's switch to macro. In some years, it is optimal for the central bank to increase base money M, to keep inflation on target, or NGDP on target, or whatever. Sometimes those changes in M will be anticipated, and sometimes they will be in response to "news". And increasing M means seigniorage revenue for the government, which owns the central bank. (See how simple macro is!)
Now let's put macro and micro together. Add that seigniorage revenue to the microeconomist's intertemporal government budget constraint. (Yes, I know that intertemporal government budget constraint will not exist if r < g always, but that isn't an optimal tax/transfer equilibrium, and I'm talking about what is optimal.)
Suppose, in the same year (or month, whatever) the central bank gets news that tells it to increase M, and the government gets news that tells it to cut A (increase the universal lump sum transfer). Though it is very very unlikely the magnitudes of the change in A and the change in M will be exactly the same, they will overlap to some extent. We could say that such a government is then doing "helicopter money". Relative to the benchmark of what was anticipated, the combined sets of news will be telling us to do money-financed universal lump-sum transfer payments.
Will those two sets of news be positively correlated? Probably yes, because anything that decreases the real interest rate would increase the current optimal transfer payment and also increase the demand for base money. But all my argument depends on is that they are not perfectly negatively correlated.
If there is zero correlation, helicopter money will be optimal 25% of the time.
If my brain were working better, I would probably talk about the effect of increased seigniorage on reducing the shadow price of public funds. But it isn't, so I won't. And it probably doesn't matter much anyway, because it's a wealth/income effect, and I think the substitution effects usually matter more. If marginal benefit curves slope dowqn, and marginal cost curves slope up, it's normally better to spend an increase in public funds on a little bit more of everything, today and tomorrow. Unless the curves shift at the same time.
I don't think there is anything here that either Scott Sumner or Rajiv Sethi would fundamentally disagree with. It's just that things get lost in translation.