Basically, dependence of Y on X means the distribution of values of Y depends on some way of the value of X. That dependence can be on the mean value of Y (the usual case presented in most of the answers) or whatever other characteristic of Y.

For example, let X be 0 or 1. If X = 0 then let Y be 0, if X= 1 let Y be -1, 0 or 1 (same probability). X and Y are uncorrelated. On mean, Y doesn't depend on X because whatever value is X, the mean of Y is 0. But clearly the distribution of values of Y depends on X value. In this case, for example, the variance of Y is 0 when X=0 and > 0 when X=1, thus there is, at least, a dependence on variance, i.e. there is a dependence.

So, linear correlation only show a type of dependence on mean (linear dependence), that in turn is only a case of dependence.

Basically, dependence of Y on X means the distribution of values of Y depends on some way of the value of X. That dependence can be on the mean value of Y (the usual case presented in most of the answers) or whatever other characteristic of Y.

For example, let X be 0 or 1. If X = 0 then let Y be 0, if X= 1 let Y be -1, 0 or 1 (same probability). X and Y are uncorrelated. On mean, Y doesn't depend on X because whatever value is X, the mean of Y is 0. But clearly the distribution of values of Y depends on X value. In this case, for example, the variance of Y is 0 when X=0 and > 0 when X=1, thus there is, at least, a dependence on variance, i.e. there is a dependence.

So, linear correlation only show a type of dependence on mean (linear dependence), that in turn is only a special case of dependence.