As Adam pointed out in his post earlier, the Deseret News story seems to make a conflict out of two findings that probably do not conflict. The seemingly different findings can be reconciled when a few basics about the nature of public opinion are understood. What are some of the basics?
First, information is quite important for having and shaping opinions. The more information in the environment about a particular policy, the more likely individuals are to have access to information.
Second, when individuals are asked specifics about policy or something else, those individuals who have the most information about that policy will be more likely to offer an opinion.
Third, the source of the information matters. Individuals have strong incentives to trust information from sources they find credible.
Finally, individuals who have relatively weak opinions about issues are the most likely to change their opinions. Conversely, people with strong opinions are least likely to change. Complicating the opinion-change calculation is that individuals with the strongest opinions are also the ones most likely to have information about policies (See John Zaller, The Nature and Origins of Mass Opinion).
Taking all of these together, we would expect some uncertainty to appear among respondents when asked about the LDS Church’s stances on several different bills (not much information about each bill). As the issue receives more coverage, those individuals who had little information at the beginning would be expected to show the most movement.
So public opinion has both a static element (what do people know at a given time) and a dynamic element (when does a change in the amount of information produce changes). Both of these elements seem to be in play with the two analyses.