It's helpful to go a bit further for these corrections. What's the reason not to use "uncorrelated" here?

In ordinary English, "uncorrelated" is indeed used for this (and a host of other things, because ordinary English is very vague). The problem is that it means something else in probability theory, namely the much weaker statement E(a-E(a)) E(b-E(b)) = E((a-E(a)(b-E(b)), which is implied by independence (p(a,b) = p(a)p(b)), but not does not imply independence. If we want to speak to those who know some probability theory, this clash of meaning is a problem. If we want to educate those who don't know probability theory to understand the literature and be able to talk with those who do know probability theory, this is also a problem.

(Note too that uncorrelatedness is only invariant under affine remappings (X and Y chosen as the coordiantes of a random point on the unit circle are uncorrelated. X^2 and Y^2 are perfectly correlated. Nor does correlated directly make any sense for non-numerical variables (though you could probably lift to the simplex and use homogeneous coordinates to get a reasonable meaning).)