In some papers, cross validation is done after train/test split via splitting the test set into folds. In other papers, cross validation is done without any other splits.
It is likely that you have encountered abuse of terminology as relates to cross validation. The truth is, most neural network papers don't actually do cross validation. In fact I have seen papers at CVPR that do model selection and performance reporting all using a single val/test set. Coming to your question:
Wikipedia article on cross validation has a pretty good summary – "One round of cross-validation involves partitioning a sample of data into complementary subsets, performing the analysis on one subset (called the training set), and validating the analysis on the other subset (called the validation set or testing set). To reduce variability, in most methods multiple rounds of cross-validation are performed using different partitions, and the validation results are combined (e.g. averaged) over the rounds to give an estimate of the model's predictive performance."
Given that repeating neural network training multiple times, each time using a different train set drawn from your dataset, can be computationally expensive, the most commonly used model selection strategy is to split the dataset once at the start of the experiment into train/val/test split. Then, you train once on the train set, while tracking performance on the validation set. Model selection is done quite simply, as you said, by picking the model that does best on validation set.
This lecture (~12 mins only) from Andrew Ng describes this very clearly.
