In a way, I think you and your parent are both right. The missing link is that the analogy is based on multivariate series. First order deals with each variable

In a way, I think you and your parent are both right. The missing link is that the analogy is based on multivariate series. First order deals with each variable in isolation. X1, X2, X3, . . . Second order contains interaction terms like X1 x X2 (The X1^2 can be thought of as a self interaction.) Third order would consider interactions between three variables and so on. I’m not sure if this a false etymology, but it’s how I’ve always understood the term and I’ve met others who understand it this way as well. I think the origin may be from linear regression with polynomial terms, which is very common in the medical sciences and social sciences. Terms like X1 x X2 are called interaction terms in that context and coefficients are often treated as quantifying the impact of a term on the result. Strictly speaking, the analogy would then be with any multivariate polynomial approximation, not just Taylor series.