Is that a definition of correlation inside the field of mathematics? Because the normal English definition of correlation just means a connection, and absolutely doesn't mean linear.
edit: Ah, Wikipedia intro suggests that yes it's specific within statistics (though still seems to not be a hard requirement to call something a correlation?): "In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it normally refers to the degree to which a pair of variables are linearly related." https://en.wikipedia.org/wiki/Correlation
The slightly more fun version of that example is that cos(x) and sin(x) have zero correlation. But if you plot a bunch of coordinates that are generated that way, you'll quickly notice a pattern...
You can easily view cos(x) as causing the value of sin(x) to change - if cos(x) starts going down, sin(x) will too soon afterward. If cos(x) starts going up, so will sin(x), soon afterward.
In fact, if you allow for a time lag between your data series, you will find that sin(x) and cos(x) are perfectly correlated.
correlation implies a linear relationship.