Poor is more of a mindset to me. Broke is simply not having capital at a given point for some reason. Poverty is a deeper condition. My parents told me we were poor and therefore ____. It was part of our identity as a family. We wore this, and drove that, and would likely go to this college and not that one (and only with government handouts) because we were poor.
I escaped that mindset and became just broke. After some work, I'm not broke anymore. And if I ever am again, I'll have the skills to not be for very long (assuming no physical or mental disability or other rare exceptions of nature).
Most poor people are also broke, but I don't think being broke necessarily makes one poor. Being poor usually keeps one broke since it is such a deeply ingrained condition. There are many components to poverty that I started recognizing early on, and they go well beyond net worth or income.