I think I define rich vs. poor mostly in terms of freedom. Very poor people are not free even from things like hunger or disease. A poor individual is tied down by jobs and debts that restrict their options: they can't take time off when they want, or quit a job they hate, for fear of losing the ability to pay rent or buy food. A "fairly rich" individual has a great deal of freedom to set their own work hours and terms, to take vacations when/where they want, to take extended time off, or to pursue a non-remunerative hobby. The truly rich are almost completely free. They can go wherever they want on a moment's notice, buy whatever they want on impulse, etc. They only need to work if they feel like it, to satisfy some desire for competition or personal fulfillment.
Can I put dollar values on these levels of rich and poor? Not really. I know people who are perfectly happy backpacking almost full time. They have no demands on their time, they can buy whatever they want (because what they want is very little) and their only significant worry seems to be the weather. They make almost no money, but I'd call them rich. At the same time I know people who make hundreds of thousands of dollars per year but who are so materialistic and so burdened by debt that I'd have to consider them poor.
If I had to define rich and poor in purely economic terms, I'd probably focus on assets plus five-year earning power. A rich person could retire in five years or less with an expectation of income over $100K indefinitely. A poor person can't retire at all because their retirement income would never match their debt. Someone in between might be able to retire, but would have only a subsistence income and/or might run out of money before their life expectancy.