Seniorage is the profit between cost of producing currency and the value of the currency.

Technically there is no seniorage when there is a loss, historically it was called an inflationary tax, meaning a loss in value to the existing money supply you hold.

I think you're mistaken -- seigniorage is the difference between the cost and the value, it doesn't need to be a profit. Negative seigniorage is still seigniorage and many such papers treat it as such, e.g.;

https://sci-hubtw.hkvisa.net/10.2307/3552184

>Negative seigniorage is still seigniorage and many such papers treat it as such

Notice how you qualify your statement as many such papers treat it as such.

Yes, you can find some people, articles that refer to a loss as seniorage, but that’s a misuse and inconsistent with the actual definition. That’s also why when they do use the term improperly they typically add “negative”, specifically because the definition of Seniorage refers specifically to revenue/profit from printing money.