IMHO, the figure of speech "n-th order" generally comes from Taylor series approximation, where n-th term is indeed expressed using n-th derivative. Also, if the series converges, the higher order terms will become smaller, thus the practice of approximating the reality up to n-th order term.

Also, it's often practical to ignore 2nd order and higher terms, because then you have a linear function - something really easy to work with. Therefore, "2nd order effects" have effectively become synonymous to "non-linear effects".