From all I can tell about what physicists mean, their use of correlation is inappropriate.

It appears that their math of correlation is correct for the math of correlation, all independent of any connections with physics.

But from all I can tell, what physics they are really talking about is lack of independence. Two random independent random variables have correlation zero. But correlation zero does not mean independent.

So, if physicists keep using correlation and find correlation zero, sorry, but there can still be dependence.

So, net, in short, for what physics is considering in the real world, they want to know if two measurements, each a random variable, are independent or not, not have zero correlation or not.