I apologize for not originally linking to the full text (PDF), which I originally did not read, but you are wrong. The study used a linear regression analysis on survey results to find relationships between (1) the dichotomous variables of underweight, overweight, or obese vs. normal weight; has smoked or smoker vs non-smoker; and poor vs. non-poor and (2) count of chronic health conditions or scores on the physical health scale of the SF-12 (PCS-12), respectively.

Nor does this imply a fat smoker is likely to come to harm from the fat rather than the cigarettes. It's about the causes of morbidity as a society not per individual.

A regression analysis models the relationship between variables. With regard to these survey results, it tells you how much a change in one variable predicts change in another variable on average in the sample. Then, by comparing the effect sizes, the researchers attempt to determine whether differences between the strengths of the relationships are significant. This study is not about how much the variables contribute to morbidity rates in society. It is about the relationship between risk factors and morbidity of various health conditions as well as physical well-being in individuals on average (but not in any particular individual).

Not a better predictor, associated with as much.

I should have said stronger rather than better, but "at least as much" and "as much" do not mean the same thing. Specifically, the study found that severe obesity has a significantly stronger effect than smoking on both measures in women and that both moderate and severe obesity have a significantly stronger effect than smoking on chronic condition count in men. There was no significant difference in effect sizes in moderate obesity vs. smoking in women on both measures and obesity vs. smoking on PCS-12 score in men, but what I said was

That said, all I can find that compares the two is this RAND study that found that severe obesity is a better predictor of a variety of chronic health problems than smoking, heavy drinking, and poverty.

One thing I got a bit wrong was that severe obesity was not a stronger predictor than poverty in men, but that wasn't the issue at hand.

And finally this goes to prove what a blight journal paywalls are on science and society.

Agreed. And I should note that it's pretty rude to accuse someone of lying about something that can easily be misinterpreted, as you have demonstrated.