# correlation and independence: common mistake?

Almost everywhere I read

If the variables are independent, Pearson's correlation coefficient is 0

I understand that correlation gives information only for linear dependence/independence variables and, if the two values are linearly independent, then their correlation is zero. Instead, the term independent is often used as statistical independence (also in https://en.wikipedia.org/wiki/Correlation_and_dependence where in correspondence of independence there is a link to statistical independence) but linear independence is not the same of statistical independence (statistical ind. $\implies$ linear ind. but not but not vice versa) so, is there a common error (also in wikipedia) or something is not clear to me?