Ah, yes that is more or less my understanding of it as well. Though I would like to see how it would perform if given the state of the board as input to predict the next move, rather than a sequence of moves, since that is how we humans normally determine the next move. I believe the move history is only relevant when it comes to en passant and certain draw scenarios (like repetition and that 50 move rule). Needless to say, it would first have to be trained on those types of inputs, which it probably is not.

Edit: move history can also be relevant when it comes to castling.