While I don't think you're wrong, they are talking about different things. One is how you're doing despite being working class. Are you a making it? Keeping your head above water? A dentist or engineer making $250k+? Do you struggle to eat or clean yourself every day?
The other is how do you make your money. Do you work for it or do you exploit for it. Were you simply born into it?
That's not a definition of middle class at all. If you live in your car you're lower / poverty class.