Richness is sustainable. If you make $10k a day, you are rich regardless where you live. If you make $10k in one day and nothing the rest of the year, you aren't rich, even on that one day.
Richness is based on what you can afford, not what you choose to have. If you have $500 million of assets and you drive a Ford and do all your shopping at Walmart, you're still rich.
In summary, you are rich when you can sustain a lifestyle requiring significantly more money than an average resident of your area.
"Significantly" and "area" are flexible and somewhat subjective.