Working and middle class by the American cultural standard (as in, defined solely by income bracket and working conditions) are ultimately useless designations meant to drive a wedge between the working class or proletariat as defined by marx; “individuals who sell their labour power for wages and who do not own the means of production”; those responsible for creating the wealth of a society.

The american concept of class is harmful because it groups together the working class members most vulnerable to exploitation; those earning the least and doing the hardest work for the longest hours; and pits them against the better off members of this very same class; those with more time, formal education, and expendable income; who might have more power and means to do legwork in organizing and uplifting the class as a whole through solidarity and collective action.

You are somewhat correct though, in that it would be difficult to make over 250k/yr without engaging in some sort of profit from capital ownership and/or labor exploitation. That person would become middle class and cease to be working class not because they made over an arbitrary dollar amount per year, however, or because they went to school in order to do so, but because they are earning profits from capital ownership and labor exploitation rather than solely those generated by their own work. Yet, by working for most of their living, they are still subject to many of the same conditions as the working class and would likely still benefit from solidarity with the working class.

via https://uregina.ca/~gingrich/o402.htm

The lower middle class or the petty (petite) bourgeoisie, “are smallowners who still work their own means of production, or owner-workers” (Adams and Sydie, p. 134). The characteristic of this class is that it does own some property, but not sufficient to have all work done by employees or workers. Members of this class must also work in order to survive, so they have a dual existence – as (small scale) property owners and as workers.

Capitalists are the owners of capital, purchasing and exploiting labour power, using the surplus value from employment of this labour power to accumulate or expand their capital. It is the ownership of capital and its use to exploit labour and expand capital that are key here. Being wealthy is, in itself, not sufficient to make one a capitalist. To be a capitalist or member of the bourgeoisie, the owner of a sum of money must be actively involved in capital accumulation, using this money to organize production, employ and exploit labour, and make the money self-expansive by using the surplus value to continue this cycle of capital accumulation.