That was the point of me bringing up multivariate models. I deal with these quite a bit.
Conceptually, you might look at, say, 50 different variables you think might help predict something due to positive, or indeed negative, correlations (you can always put a negative sign on a variable, so negative correlations are just as useful).
After running your regressions, you might end up including 25 of them in a multivariate model. What you are finding is the 25 you excluded, while individually correlated, actually reduce the accuracy of your model. This can happen when the 25 you are including basically already contain almost all the predictive information in the remaining 25, and there is noise in those remaining 25, and so including them would actually add more noise and very little useful information.
OK, then someone proposes a new variable for the model, so you test what happens when you include it. Maybe you will find it helps the model to add it. Maybe you will find it hurts. Maybe it helps AND it knocks out some other variables. Maybe it helps AND it knocks out some other variables AND now some of the 25 you originally excluded are back in. And so on.
These things are very tricky and different people can approach the same sort of question and come up with very different models just based on relatively subtle differences in data, what they are seeking to predict, what other variables they consider, and so on. So, one model might include a variable, and another model might exclude it.
OK, so SAT on its own is weakly correlated with first year GPA at Dartmouth. Just to begin with, other colleges might not think that first year GPA is what is important to predict. And then of course they have different internal data, they may consider different other variables, and so on.
But then Darmouth finds including SAT on a long list of variables in a multivariate model helps its model, so it includes it. Yale finds any of SAT, ACT, IB, or AP help its model, but not all are necessarily, so it goes text flexible. Some other college finds SAT does not help its model, so it excludes it. And so on.
I know this is a bit long already, but I think a lot of confusion in these conversations arises from not really understanding how complicated, and sometimes counterintuitive, the choice of which variables to include in a multivariate model can end up being. And it is not at all a surprise to me different colleges would come to different answers.