I think that depends on context. Sometimes, in a technical setting correlation just means dependence as an abstract concept, and this includes non-linear dependence. Similar how in financial circles, volatility doesn't mean standard deviation, but in colloquial settings it does.
Correlation, in general, just means some sort of statistical dependence: knowing x tells you something about y. It's often "operationalized" by computing Pearson's r: it's easy to do and there's lots of associated theory.
However, I would find it absolutely bizarre if someone showed a plot with obvious non-linear dependence and described it as "uncorrelated". In that case, the low r reflects a failure of the measuring tool rather than something being measured.
