The rich already pay different amounts than the poor for goods and services in pre-tax dollars. That's kind of the idea, IMO. If my average tax rate is 28% and a poor person pays an average 5% then the same $5 SF coffee costs me $6.94 and them $5.26.
The dictionary definition is quite clear: A progressive tax is a tax in which the average tax rate (taxes paid ÷ personal income) increases as the taxable amount increases. [Google]
A regressive tax is a tax applied uniformly, taking a larger percentage of income from low-income earners than from high-income earners. It is in opposition to a progressive tax, which takes a larger percentage from high-income earners. [Also Google]
As such, Medicare taxes are regressive because they are capped at some amount of earned income, whereas the ordinary income tax rate is progressive because your marginal rate increases as you earn more money.
You are correct that a flat tax is neither progressive nor regressive with respect to dollars, but it is regressive with respect to marginal utility of those dollars to the earner. "Fair" is a value judgement and colors the conversation.