If you are making $250k/year through a job, and you depend on that money for daily life, you are not wealthy. You are a high-income earner. You can kind-of put your kids through private school and maybe even get a rent-a-maid more often than lower-earning folks, but if your yearly spend is close to $250k/year, you're not wealthy.
If you are making $25M/year through a job, and you depend on that money for daily life, you are still not wealthy. You'll be classified as an ultra high net worth individual by banks and money managers and qualify for a lot of cool rich people shit. You'll probably trade American or Delta for NetJets or Delta Private Planes (a benchmark a lot of folks in this article used as "elite snobbery," which it really isn't in many cases), or you'll make the mistake of buying your own jet, but if your yearly spend is >$25M/year, you're not wealthy.
Why? If you lose your job in both cases, you are screwed.
Wealthy is never having to work for your money because your money works for you. Having a high-income helps towards getting to wealthy, but you can also just buy more expensive shit with that income and just stay high-income.
The way I see it: you're wealthy when you can do, wear or say nearly whatever the fuck you want and not worry about your money/lifestyle going away.