I thought progressive and regressive referred to convex and concave taxation curves, respectively (both monotonically increasing, of course). Then flat tax is exactly the border (i.e. a linear function).
While depending on background, that may be what some people think of as defining progressive and regressive taxes, and may indeed describe at a low level the concept that most people think of and even be the origination of the term, it's not the definition generally put forth currently[1][2] (which is, admittedly, very simplistic). That just goes to my point, which is that people aren't even necessarily in agreement on the terms, and might not even be aware of that.
I think these definitions are equivalent. If your marginal tax rate (at least sometimes) increases as you get more income, the curve of tax liability against income will be concave upwards. If it (at least sometimes) decreases, that curve will be concave downwards. Right?
Almost. I think the concept inside many people's heads is simplified to the degree that it doesn't even need to describe a curve. A simple step function, for example (the simplest in this case being if you make less than $X, you pay nothing, otherwise Y%). In that manner, the concept is not approximating the curve, the curve is approximating the concept.
1: https://apps.irs.gov/app/understandingTaxes/student/glossary...
2: https://apps.irs.gov/app/understandingTaxes/student/whys_thm...