My definitition or criteria for someone being wealthy would be having enough money for a person with average intelligence and discipline to live off of just the earnings from interest, dividends, and capital gains without having to work if they wanted to. These days with a constantly increasing cost of living, I would say this would be around at least $10 million.
Some may consider this really high. The way I think about it though, being wealthy by definition means you have a high degree of financial security (though nobody is ever completely financially secure, even billionares can lose everything), meaning that they have a margin of safety against making financial mistakes, market volatility, and bad luck (major health problems, family issues, being sued, etc.) to the point that they would still be secure if these things happened. Also with inflation moving up so fast like it is now, my figure accounts for this natural erosion of purchasing power.
Then again I myself would just be satisfied with an extr $5,000 :)