The fight is about the predictor language in some cases. Because it's only a trivial difference to those who don't understand the details of how these things are made. In pre-training the model really is trained to predict the next token. What is being emitted by the model is, by structure, by training and by optimization, a prediction of the very next token.
What is emitted by a model during RLHF and RLVR is not, by structure, training or optimization, a prediction of the next token.
What is emitted by a model during RLHF and RLVR is not, by structure, training or optimization, a prediction of the next token.