Well, it depends on how the person with the money is viewing "rich". If you mean they think of themselves as along the lines of Trump, probably multi millions, but if you mean after I'm done paying the bills on things and property I don't necessarily have to have while having spare cash left over each month, then that's me.
I mean, I wouldn't snub my nose at multi millions of dollars, but I honestly don't want to work that hard for that long and then have to sustain it and pay others to help me manage it. Contrary to popular opinion, most people in the U.S. don't inherit large sums, they work for it.
I'd want a nice house in a nice neighborhood. Something with ample space and an extra room if I should ever need to help a friend or family member (that's of course on top of extra rooms like: one for storage, one for exercising, and one for the washer/dryer). A good one like described above would be rented in my county for about $2,000 a month.
Then there's auto insurance, costs for automobile repair and maintenance, food, utility, personal care, and the joys of life like goodies and presents to me. And then some basic upkeep like lawn mowing, but not much, I like to handle things myself. Then dreaded taxes, Fedzilla, state, and local.
I'd say $5,000 to $7,000 a month before taxes would make me rich. Enough left over, if need be, to help a friend of family member in the short run. Any more than that and I'll start coming up with ways to use the money that start to get involved.
Now poor is another story. The definition of poverty here in the U.S., is quite different from other countries. Many poor people or ones described as being below poverty, still somehow manage to have a cellphone, a computer, a TV, and feed themselves each month, and not necessarily all those above for each person.
There are two kinds of poor that I wouldn't want to be and I've already been one:
The homeless kind where your address is where ever you park your car; you struggle with food, gas, and may or may not have some kind of job. The prospect of getting out of it one day can be a reality. While not everybody is James Cameron, he did write "The Terminator" while living in his car.
Then there's so poor there's potentially no hope. The kind where you sleep in the woods (and not by personal choice), walk (or bike if lucky); you're worn, hungry, not in good health and you might try to get ahead but nobody wants to hire a guy or girl without a physical address or one that can't come to work clean. Plus they are suspicious of why you're in that situation.
The last one, to me, would be poor. I wouldn't want to be there. It could be worse, but that's bad enough.