Not the OP, but poor as used here seems to refer to average quality of life , quality of infrastructure, etc.

> Is the result of that a higher median income, or is it a reflection of a higher wealth inequality?

Higher wealth inequality leading to stretched public services and infrastructure, which lead to lower quality of life , despite higher nominal GDP per capita.

You are probably much better off being a poor person in Spain (33k GDP/capita) vs Mississippi (40k GDP per capita), because at least you don't need to worry about the cost of healthcare.

You're more likely (but still very unlikely) to get extremely rich in the US though, although probably not in Mississippi.