On the contrary, I'm becoming even more convinced that "fat" is an indistinct and malleable term the more I think about it.
You state that since we have metrics in place, for things like fatness and poverty, that it's all we need, even though you acknowledge that these metrics are subject to error. The point the other poster has made is that the metrics may not be just wrong, but also unclear. Whichever is the lesser of the two evils, the greatest evil is obviously to be both wrong and unclear.
Starting with your example of poverty. What is it? Is it not being able to afford things necessary to survival? Or is it being permanently lodged in a certain social class, i.e. the poor? Or is it having to struggle financially? Within even just these three examples, you have the numerous debates about things like what is enough to survive, what separates one social class from another, and how much hardship is enough to be struggling. The answer to these questions will determine a lot, like who can enter what program, who gets what aid, what taxes are appropriate for whom and so forth.
Regardless of how long you have admitted to being overweight, the BMI is famously unreliable. It cannot factor in healthy weight, such as skeletal muscle or bone density, and compare it to unhealthy weight, generally adipose tissue. It can't even account for primitive measures like somatotypes. Competing metrics drastically change the amount of people considered overweight by the BMI. A WHR measure, for instance, would increase the amount of people considered overweight two to three times what are considered such under a BMI.
You have a much more convincing point that people who are overweight recognize it already, metrics be damned. You may be right or wrong about that, but, as it is, the study in question generally tends to disagree with you, implying that fat peers will alter one's understanding of fatness, or at least what's acceptable. It doesn't directly refute you, but it does suggest a high degree of malleability in regards to what one considers fat.