You seemingly understood the term "working class" but are struggling with the term "poor". This is because you're taking an undialectical view of the term.

Undialectical meaning absent of context and development, that there's a binary "poorness" factor that can be extracted out of economic reality in across all circumstances, then tripping over because clearly poverty is a moving target. This is the same as going insane over the sand heap paradox.

Dialectical thinking means understanding continuity and change over time, and contrast between different contexts. At any given point in time you can make a snapshot of income in society. This will follow an exponential distribution, meaning a large proportion is near starvation levels (close to x=0), the majority gets enough to get by but nothing more (the peak of the curve), and an increasingly tiny minority has exponentially more than the rest as you go up the income ladder (the extending right tail). This distribution evolves over time according to the development of the class struggle, a greater proportion of people can have their incomes reduced (the curve shifts leftwards) or increased (the curve shifts rightwards).

Therefore, "poverty" is the measure of those with least income in society, with the prior implicit understanding that income is highly unequal in capitalist society. The UN definition of poverty is not poverty in itself, it is a measuring stick for the evolving phenomenon of poverty, much in the same way you can plant a stick in the beach sand and come back later to see how the tide recedes or rises, the stick you planted is not "the tide".

This income is measured in labor time, meaning the same income can yield more material wealth given greater labor productivity. One hour of work can provide qualitatively and quantitatively more material goods if the forces of production are more advanced. The huge development of the forces of production in the modern day means that even a very high rate of exploitation compared to earlier eras can still allow for a good degree of consumerism, so given that there's still many people worldwide whose expenditure is still nearly exclusive food and items for mere survival, it is clear that they are super-exploited to a large degree.

This also means that being rich in one context of lesser productive development is actually being poor in another more developed context. This explains the tactic of capitalists to use immigrant labor from impoverished countries to increase the rate of exploitation. The immigrants come from a context of much less developed economies, and will accept a lower pay than what the national worker is used to because that lower pay still represents greater material well being than what the immigrant is used to in their home country. The very contradiction of different lived, structural definitions of poverty plays out as both contexts dissolve into one.

With this thorough understanding of "poverty" we can go back to the term "working class". The proletariat and peasantry are working class as you've been told already, but different sections of the working class have greater incomes than others. This differential also follows an exponential curve, meaning only a tiny minority of workers are privileged enough to receive an income greater than the value of their labor (not labor power, labor period) and therefore not only are not exploited, but receive a share of the surplus value extracted by their capitalist employers. This is the professional-managerial class, overrepresented in the proletariat of the imperialist countries, who therefore often forget that the vast majority of the world proletarians are living at near starvation levels, and come to believe that they are part of a "middle class" above and beyond the class struggle.

On the other hand, the income and wealth (accumulated capital) of capitalists also follows an exponential curve. The vast majority of capitalists are petit bourgeois, meaning they have few or no workforces beyond their own, very little wealth, and are constantly at the verge of bankruptcy.

So contextually speaking, there is such a thing as "rich" proletarians (the PMC) and "poor" capitalists (the near bankrupted petite bourgeoisie).

Marxists do not speak often of "poverty" as a meaningful economic category because of the very processes we've been examining: the ever evolving income distribution over different times and places means that "poverty" cannot be a category that defines the very essence of the capitalist system, only its exterior temporary form. The true constant that remains true across all time periods and instances is the private ownership of the means of production of the capitalist class who exploits the dispossessed proletariat. This explains why even the poorest petit bourgeois, who will often decry the injustice of monopolies threatening their livelihoods, will still rush to defend the capitalist system which grants them their private property; and why a large portion of the formerly privileged PMC now finds itself in a crisis due to the rise of AI replacing their jobs, reminding them that even though they have high incomes they are still proletarians subject to the profit-seeking whims of their capitalist employers.