How is a test task being nonparametric or parametric defined?
My understanding is that the test task is parametric, if and only if it assumes a parametric model on the distribution of the sample, regardless of whether its null is true or not. For example, the test task that t tests solve assumes the sample is normally distributed, regardless of whether their nulls are true.
But what if the distribution of the sample isn't assumed to be parametric, but the null specifies the distribution of the sample being parametric? For example,
when the null specifies that the distribution of the sample is normal, i.e. testing normality of the data. Is the test task parametric or nonparametric?
when the null specifies that the sample has a specific distribution, i.e. goodness-of-fit test, is the testing task parametric or nonparametric? Such as the test tasks that the chi-square test solves, and the test task that the Kolmogorov-Smirnov one-sample test solves?