The correlation measures the amount of interaction between two variables, so you’d want to use this when figuring out diversification benefits, co-movements, etc. In most problems, they’re just making the assumption that the r-squared is relatively high. Don’t overthink this…
One thing I’m confused about: if Correl (1,2) = Cov(1,2) / (Stdev1*Stdev2), then how does that fit with the reading on international diversification that says higher vol (ie stdev) causes the correlation to increase (v3, p369)? Shouldn’t it decrease based on the formula, or can you not think about that concept in mathematical terms?