That's a fair point and I understand why you'd argue that.
However, you'll notice that most people consider rich to be "having more than what I have" or "having more than other people". Many millionaires don't think themselves wealthy -- and they're not necessarily wrong. They've reached a point where many of their peers have more capital than they do. Others may consider them "poor".
That's a great definition of being middle class. I'd argue being rich involves having far more resources than that.