The new paper confirms the current mainstream view that birth order can only account for very small differences in personality in Western countries. They have added a result on differences in honesty/humility, a trait that is less studied than the other traits in the study. 

They also studied the idea that differences in family size/sibship size cause differences in personality. They interpret their findings in support of this notion and it is here they discuss differences to prior studies. However, their analysis strategy fails to adjust for the many, many factors that are different between families other than the number of children. They adjust only for religiosity, but not for country of residence or birth, whether English was the mother tongue, socioeconomic status, the personality of the parents who choose to have large families, and so on and so forth. Their finding that children of larger families tend to be more honest and more agreeable could thus result from the family environment, as they infer. Or it could be better explained by a third variable, for example, it could just reflect higher cooperativeness of some geographic region at a particular time, if this region had higher fertility. This might explain differences with previous studies, which usually were conducted within a single country.