A move as in say "move 16 or move 5" describes both white and black taking their turn. A half move is a move by either black or white. So - "half moves since last capture" - is precise. But as to why this information is considered useful? Maybe it's something to do with the rules regarding drawn endings as in "50 moves since last pawn move or capture"...although it's never happened to me and I never quite manage to remember the rule properly.