When you explore a node in alpha-beta, there are basically three possible results:
- You find a move whose score is beta or better. In this case you can stop considering moves and return the score you found. You store the score as a lower bound in the TT and you also store the move that produced the beta cut.
- All your moves have a score that is less than beta, but at least one of them has a score above alpha. In this case you store the score in the TT as an exact score and you also store the move that gave you the best score.
- All your moves have scores that are alpha or worse. In this case you store the best score you find as an upper bound, and now whether you store the best move in the TT or not is not very important, in my experience. Moreland doesn't store it. I do. Moreland's position probably makes a little more sense, but in practice my program performed about the same with or without storing the move.

Moreland's pseudo-code is not very clear, but it shouldn't be hard for you to figure out how to keep track of the best move. If you need to see some pseudo-code, I can produce some.