They compare the degree of correlation between their identical and non-identical twins. If in both groups, twins show on average the same level of procrastination/whatever trait, you can infer that the trait is not strongly genetically linked. If the identical twin group shows more correlation between siblings than the non-identical twin group, then you can infer that the trait is strongly genetically linked. You can read more under the first few paragraphs of their data analysis section.

Just thinking out loud here, but if identical twins were much closer in, for example, appearance, couldn't this induce more similarity in other traits. That is, if being identical made twins have closer relationships, might that not result in higher correlations in certain traits, which weren't due to genetics.

Do people tend to know if they are identical or non-identical twins?

Sure! If one is a guy and the other a girl, they would guess pretty quick.

Just kidding. But in my experience, parents of identical twins go in for the "isn't it cute lets dress them the same and treat them the same and tell them they are just the same". That has to be a strong influence. I'm not sure the twins-identical-or-not goes near far enough to untangle the nurture question.

That may well be true in general - twin studies are certainly not the end all of distangling genetic and environmental effects. But if you read the studies results, I think the difference in correlation effect size speaks for itself. In general, identical twins correlate with r value of ~0.5 and non-identical twins with <0.1.

The authors interpret the low correlation in non-identical twins as evidence that the traits being studied are weakly (if at all) influenced by environmental factors that non-identical twins are subjected to. They then infer (assert, w/e) that this level of weak environmental influence also extends to the set of environmental factors that identical twins are subjected to. Now, obviously as you point out, there are environmental factors that are more or less unique to identical twins, but I feel that its unlikely (and so did the authors) that these identical twin specific factors contribute significantly the vast difference in correlation between the two classes.

It could just be that they underestimate the degree that identical twins are driven to duplicate one another's mannerisms. Did they try to measure it? Or just dismiss it.