I think you may be misunderstanding my objection, which has nothing to do with any notion of purity.
Suppose you measure the “logini coefficient” in some country. Immediately afterwards, they decide that instead of denominating their currency in Foobars, which is beginning to produce slightly silly results after many years of inflation, they will switch so that the New Foobar equals 1000 Old Foobars.
Suddenly everyone’s income is nominally 1000x less. All your logs will decrease by whatever log-to-your-chosen-base of 1000 is. And this will completely change the “logini coefficient”. Even though nothing has really changed.
There may well be a good answer to this objection. Perhaps there’s a good case to be made for taking the unit of money to be, say, the price of a mass-produced loaf of bread or something. (The “good case” might be something like “meh, there’s no particular justification for this, but it produces results that seem like they make sense”.) But prima facie, a metric whose value can change drastically because of a change that makes literally no real difference seems problematic.
I think that most macro indicators will be vulnerable to random changes in currency denomination, which is why (as you suggested) economists don’t look at just nominal value. If you’re tracking changes in Logini from 2007-2017 you could convert everything to 2007 Foobars, or you could use the “loaf of bread” approach and adjust for Purchasing Power Parity. PPP is the standard way to compare currencies across time and space, it’s not perfect but it gives reasonable results.
Is there any reason to interpret the coefficient by itself, rather than relative to other coefficients of other areas/times? If you were comparing coefficients you would of course use the same real-adjusted currency. If you aren’t comparing, I don’t see the point of the coefficient nor the problem with it changing due to the income units used.
I’d have thought you’d want to be able to say things like “if the coefficient is at least 0.618033989 then you should give very serious thought to whether you have a problem”.
And it’s all very well to say that if you were comparing then of course you would use a common currency for both—but in practice I doubt it would work that way. If you compare two countries’ GDPs, you look them both up, do a single currency conversion, and compare the numbers. You can’t do that with logini; if you know that a country’s logini is 0.739085133 when computed using pounds sterling in 2017, that is not enough information to convert it into US dollars in 2020 (even in 2020 when you know how much the dollar is worth then).
I suppose that if logini coefficients took off then people computing them might pick some standard-for-all-time currency and convert to that. Which is more or less equivalent to my suggestion of using something like the price of a loaf of bread as a unit, except a bit more arbitrary.