Regarding demonstrated interest, the specific numbers were as listed below this post. 1/3 of surveyed colleges said demonstrated interest had no significant influence on admission decisions, and the clear majority said it had either no influence or limited influence. In contrast. the clear majority of colleges said counselor LORs had at least moderate importance in admission decisions, and only 18% said counselor LORs had no influence. Teacher LORs were similar.
I think most would interpret this as colleges as a whole treat LORs as more influential than demonstrated interest (conflicting with article quote). I think the more important distinction is that there are a huge number of exceptions, with many different admission systems at different colleges, rather than a single weighting system. Some colleges really emphasize demonstrated interest in admission decisions. Some colleges do not consider demonstrated interest at all. Looking at the regression results in the appendix, lower acceptance rate colleges were more likely to say they emphasized LORs, but were slightly less likely to say they emphasized demonstrated interest. The focus of this forum tends to be lower acceptance rate colleges, so among colleges frequently discussed on this forum, one would expect a different pattern than the overall averages.
Demonstrated Interest
32% No Importance
28% Limited Importance
24%Moderate Importance
16% Considerable Importance
Comparing to Counselor LOR
18% No Importance
27% Limited Importance
40% Moderate Importance
15% Considerable Importance