NOTES: Each point represents a country.

That graph doesn't rank rich people vs poor people; it ranks rich countries vs poor countries. Lots of other factors determine global fertility rate.

As a relevant example, there's a strong correlation between the educational level attained by women and their fertility rate.

Here's the same chart from that page, in case the linked site has issues:

But that's besides the point illustrated by this post, as the post article only pertains to one country - the United States. People in the US are experiencing changes that downgrade their quality of life over time. The economy was different when we were kids, and as a millennial, my generation has pretty much only seen things get worse.

Poor people in under-developed nations have it bad, but when that's all people have known, normal life goes on. Some may even have hope that perhaps some day they can see their country grow and prosper, that they can see their children in a better world.

By contrast, the US is experiencing an economic downslide. We don't see hope on the horizon. We were raised on promises (shout out to Tom Petty), but then matured into a country that did nothing but break them. Things seem to be getting worse all the time, right in the prime of our child-bearing/child-rearing years. We have no idea what anything is supposed to cost anymore, attaining affordable housing requires winning a literal lottery, and have you gone to the grocery store lately? Buying enough for one person is already absurd, I can't imagine having more mouths to feed.

Consider this the coincidence of women having high education + low income. More of us are aware of and have access to birth control (except for many in republican states, those poor women.) It seems that right now, the influence of education is stronger than the (supposed) pro-fertility influence of poverty you claim.

Though those certainly aren't the only two measures we could use to link wealth and fertility. Religion, secular cultural influences and practices, accessibility of contraception... it's a big picture, with lots of detail to look at.