TMwR calls the elements of the initial split "training" and "testing" set and the elements of the a resample "analysis" and "assessment" set.

In rsample, both of those types of splits are rsplit objects and all of training()/ testing() / analysis()/assessment() work on all rsplit objects. We should be more consistent with that conceptual difference and make training()/ testing() work only on initial splits and analysis()/assessment() only on splits in resamples.
library(rsample)
split_init <- initial_split(mtcars)
cars_train <- training(split_init)
cars_analysis <- analysis(split_init) # this should error
folds <- vfold_cv(mtcars, v = 3)
fold_1 <- folds$splits[[1]]
analysis_1 <- analysis(fold_1)
train_1 <- training(fold_1) # this should error
Created on 2022-12-06 with reprex v2.0.2